Analytical model of the mean velocity distribution in an open channel with double-layered rigid vegetation

An analytical model for predicting the vertical distribution of mean streamwise velocity in an open channel with double-layered rigid vegetation is proposed. The double-layered model was constructed in a laboratory flume with an array of steel cylinders of two heights. For each vegetation layer (i.e., the short- or tall-vegetation layer), the flow is vertically separated into a lower vegetation zone and an upper vegetation zone, and corresponding momentum equations for each zone are formulated. For the lower vegetation zone, a uniform velocity was adopted since turbulent shear is relatively small and the Reynolds stress is ignored. For the upper vegetation zone, a power series was used to solve the momentum equations. For the free-water zone, a new expression was suggested to obtain a zero velocity gradient at the water surface instead of the traditional logarithmic velocity distribution. Good agreement between the analytical predictions and experimental data demonstrated the validity of the model.     

Numerical modeling of free surface flow over submerged and highly flexible vegetation

Vegetation in rivers, estuaries and coastal areas is often submerged and highly flexible. The study of its interaction with the ambient flow environment is important for the determination of the discharge capacity, morphological characteristics and ecological conditions of the water course where it grows. In this work the hydrodynamics of submerged flexible vegetation with or without foliage is investigated by using a 3D numerical model. Flexible vegetation is modeled by momentum sink terms, with the velocity-dependent stem height determined by a large deflection analysis which is more accurate than the previously used small deflection analysis. The effect of foliage on flow resistance is expressed in terms of the change in the product of the drag coefficient and the projected area, which is supported by available experimental data. The computed results show that the vertical profiles of the mean horizontal velocity and the vertical Reynolds shear stress are correctly simulated. The temporal variation of the stem deflection follows closely that of the velocity and the ‘Honami’ phenomenon can be reproduced. The numerical simulations also confirm that the flexibility of vegetation decreases both the vegetation-induced flow resistance force and the vertical Reynolds shear stress, while the presence of foliage further enhances these reduction effects.     

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.     

Power-law index for velocity profiles in open channel flows

Turbulence theory has demonstrated that the log law is one of the established theoretical results for describing velocity profiles, which is in principle applicable for the near-bed overlap region, being less than about 20% of the flow depth. In comparison, the power law that is often presented in an empirical fashion could apply to larger fraction of the flow domain. However, limited information is available for evaluating the power-law exponent or index. This paper attempts to show that the power law can be derived as a first-order approximation to the log law, and its power-law index is computed as a function of the Reynolds number as well as the relative roughness height. The result obtained also coincides with the fact that the one-sixth power included in the Manning equation is of prevalent acceptance, while higher indexes would be required for flows over very rough boundaries.     

A mixing-length model for predicting vertical velocity distribution in flows through emergent vegetation

Abstract

The vertical profiles of streamwise velocities are computed on flood plains vegetated with trees. The calculations were made based on a newly developed one-dimensional model, taking into account the relevant forces acting on the volumetric element surrounding the considered vegetation elements. A modified mixing length concept was used in the model. An important by-product of the model is the method for evaluating the friction velocities, and consequently bed shear stresses, in a vegetated channel. The model results were compared with the relevant experimental results obtained in a laboratory flume in which flood plains were covered by simulated vegetation.     

Velocity distribution in a gradually accelerating free surface flow

This study investigates the interaction of the vertical velocity v and the streamwise velocity u in a gradually accelerating flow. The analytical result shows that the momentum of uv driven by the mean velocities in a non-uniform flow is not negligible. This additional momentum directly results in the concave profiles of Reynolds shear stress in gradually accelerating flows, a departure from the expected linear profile. Consequently, this momentum causes the maximum velocity to be located below the free surface, i.e., the dip-phenomenon. This paper investigated the interactions of the Reynolds shear stress, non-zero vertical velocity and dip-phenomenon, it is found that the non-zero vertical velocity causes the dip-phenomenon. The approach is tested using the experimental data of Song and others, and good agreements between the predicted and measured velocity profiles have been achieved.     

Shear velocity estimates in rough‐bed open‐channel flow

Shear velocity u_* is an important parameter in geophysical flows, in particular with respect to sediment transport dynamics. In this study, we investigate the feasibility of applying five standard methods [the logarithmic mean velocity profile, the Reynolds stress profile, the turbulent kinetic energy (TKE) profile, the wall similarity and spectral methods] that were initially developed to estimate shear velocity in smooth bed flow to turbulent flow over a loose bed of coarse gravel (D₅₀ = 1·5 cm) under sub‐threshold conditions. The analysis is based on quasi‐instantaneous three‐dimensional (3D) full depth velocity profiles with high spatial and temporal resolution that were measured with an Acoustic Doppler Velocity Profiler (ADVP) in an open channel. The results of the analysis confirm the importance of detailed velocity profile measurements for the determination of shear velocity in rough‐bed flows. Results from all methods fall into a range of ± 20% variability and no systematic trend between methods was observed. Local and temporal variation in the loose bed roughness may contribute to the variability of the logarithmic profile method results. Estimates obtained from the TKE and Reynolds stress methods reasonably agree. Most results from the wall similarity method are within 10% of those obtained by the TKE and Reynolds stress methods. The spectral method was difficult to use since the spectral energy of the vertical velocity component strongly increased with distance from the bed in the inner layer. This made the choice of the reference level problematic. Mean shear stress for all experiments follows a quadratic relationship with the mean velocity in the flow. The wall similarity method appears to be a promising tool for estimating shear velocity under rough‐bed flow conditions and in field studies where other methods may be difficult to apply. This method allows for the determination of u_* from a single point measurement at one level in the intermediate range (0·3 < h < 0·6). Copyright © 2013 John Wiley & Sons, Ltd.     

3D Numerical modelling of flow divisions at open channel junctions with or without vegetation

The flow division at an open channel junction is affected by the inflow discharge and the downstream water depths of the junction. The growth of vegetation in a channel system is environmental friendly, but its effect on the flow in an open channel junction can be significant. In this work a 3D RANS (Reynolds Averaged Navier–Stokes equation) model has been implemented to investigate the flow phenomena in channel junctions with or without vegetation. The model is first validated by two cases: flow in an open channel T-junction without vegetation, and flow in a single open channel with vegetation. The model is then applied to simulate flow in an open channel T-junction with varying width ratio and vegetation density of the branch channel. The results quantitatively predict the trend of increasing flow in the branch channel with the increase in branch channel width and/or the decrease in vegetation density. The overall energy loss coefficient of the system, however, decreases with the amount of flow in the branch channel.     

The mechanism of energy loss and transition in a flow with submerged vegetation

The mechanism of energy balance in an open-channel flow with submerged vegetation was investigated. The energy borrowed from the local flow, energy spending caused by vegetation drag and flow resistance, and energy transition along the water depth were calculated on the basis of the computational results of velocity and Reynolds stress. Further analysis showed that the energy spending in a cross-section was a maximum around the top of the vegetation, and its value decreased progressively until reaching zero at the flume bed or water surface. The energy borrowed from the local flow in the vegetated region could not provide for spending; therefore, surplus borrowed energy in the non-vegetated region was transmitted to the vegetated region. In addition, the total energy transition in the cross-section was zero; therefore, the total energy borrowed from the flow balanced the energy loss in the whole cross-section. At the same time, we found that there were three effects of vegetation on the flow: turbulence restriction due to vegetation, turbulence source due to vegetation and energy transference due to vegetation, where the second effect was the strongest one.     

Time-mean structure of secondary flows in open channel with longitudinal bedforms

Secondary motions are commonly present in open channel flows. This study aims to investigate, both experimentally and analytically, time-mean characteristics of cellular secondary flows generated by longitudinal bedforms. Experiments were conducted in a tilting, rectangular flume with six different longitudinal bedforms, including alternate bed strips with different roughness heights and bed ridges of wavy and rectangular shapes. Various flows were sampled using a two-dimensional laser Doppler anemometer (LDA) and a one-dimensional ultrasonic Doppler velocimeter (UDV). Experimental results demonstrate secondary flows appearing basically in cellular fashion over the modeled longitudinal bedforms. It is also shown that the cellular structures can be described analytically with kinematic considerations. The discrepancies between theoretical and measurement results are discussed. An empirical relationship between maximum vertical velocity and bed configuration is finally proposed based on the experimental data.     

Analytic stage–discharge formulae for flow in straight trapezoidal open channels

Analytic stage–discharge formulae are derived for flow in straight trapezoidal channels, based on the 2D analytic velocity distribution in open channels given by Shiono and Knight [Shiono K, Knight DW, Turbulent open-channel flows with variable depth across the channel. J Fluid Mech 1991;222:617–46]. A simple hand-calculation method is provided. Legendre incomplete elliptic integrals of the first and second kinds and a binomial series expansion are used in the derivation of these analytic formulae, together with physically based hydraulic parameters, such as local friction factor (f), dimensionless eddy viscosity (λ) and secondary flow (Γ). The stage–discharge results obtained from the formulae are shown to be in good agreement with experimental data, as are the corresponding analytic velocity and boundary shear stress distributions. The influences of f, λ and Γ on the stage–discharge relationship are also discussed.     

Neural network modelling for mean velocity and turbulence intensities of steep channel flows

The main purpose of this study is to evaluate the potential of simulating the profiles of the mean velocity and turbulence intensities for the steep open channel flows over a smooth boundary using artificial neural networks. In a laboratory flume, turbulent flow conditions were measured using a fibre‐optic laser doppler velocimeter (FLDV). One thousand and sixty‐four data sets were collected for different slopes and aspect ratios at different locations. These data sets were randomly split into two subsets, i.e. training and validation sets. The multi‐layer functional link network (MFLN) was used to construct the simulation model based on the training data. The constructed MFLN models can almost perfectly simulate the velocity profile and turbulence intensity. The values of correlation coefficient (γ) are close to one and the values of root mean square error (RMSE) are close to zero in all conditions. The results demonstrate that the MFLN can precisely simulate the velocity profiles, while the log law and Reynolds stress model (RSM) are less effective when used to simulate the velocity profiles close to the side wall. The simulated longitudinal turbulence intensities yielded by the MFLN were also fairly consistent with the measured data, while the simulated vertical turbulence intensities by the RSM were not consistent with the measured data. Copyright © 2007 John Wiley & Sons, Ltd.     

On the study of control effectiveness and computational efficiency of reduced Saint-Venant model in model predictive control of open channel flow

Model predictive control (MPC) of open channel flow is becoming an important tool in water management. The complexity of the prediction model has a large influence on the MPC application in terms of control effectiveness and computational efficiency. The Saint-Venant equations, called SV model in this paper, and the Integrator Delay (ID) model are either accurate but computationally costly, or simple but restricted to allowed flow changes. In this paper, a reduced Saint-Venant (RSV) model is developed through a model reduction technique, Proper Orthogonal Decomposition (POD), on the SV equations. The RSV model keeps the main flow dynamics and functions over a large flow range but is easier to implement in MPC. In the test case of a modeled canal reach, the number of states and disturbances in the RSV model is about 45 and 16 times less than the SV model, respectively. The computational time of MPC with the RSV model is significantly reduced, while the controller remains effective. Thus, the RSV model is a promising means to balance the control effectiveness and computational efficiency.