Flow Resistance over Mobile Bed in an Open-Channel Flow

This paper deals with the underlying mechanism of flow resistance in an alluvial channel: The effects of sidewall and bed form on flow resistance, Einstein’s divided hydraulic radius approach and Engelund’s energy slope division approach are reexamined. These two approaches assume that the shear stress on a mobile bed is the summation of shear stresses caused by skin friction and bed form. Using a different approach, this paper presents a theoretical relationship between the total bed shear stress with grain and bed-form shear stresses. The contribution of sidewall on the total bed shear stress is also discussed. The writers found that the size of bed form plays a significant role for the flow resistance, and developed relevant expressions for the length of the separation zone behind the bed forms. In addition, a systematical approach has been developed to compute the flow velocity in an alluvial channel. This approach is tested and verified against 5,989 flume and field measurements. The computed and measured discharge/velocity are in good agreement and 83.0% of all data sets fall within the ±20% error band.     

Flume for Teaching Spatially Varied Open-Channel Flow

Spatially varied flow in open channels is a topic that is often included in undergraduate open channel hydraulics courses. Physical and computational models are developed to enhance the presentation of spatially varied flow to engineering students at the late undergraduate or early graduate level. The physical model is inexpensive and easy to build and the computational model is easily developed using commercially available spreadsheet software. The physical model consists of a 30.48 cm nominal-diameter PVC pipe that is 6.1 m in length and has circular orifices approximately 1.40 cm in diameter drilled on 15.24 cm centers along the pipe invert. A relationship between the orifice discharge coefficient and a modification of the Froude number, as measured in the flume upstream of the orifice in question, was developed in repeated trials having varying flume slope, volumetric inflow rate, and end conditions. With this relationship, a stepwise solution to the energy equation is used to predict the water surface profile. Differences between the water surface profiles observed and predicted in repeated trials averaged approximately 2 mm.     

Numerical Simulation of Flow over a Rough Bed

This paper presents results of a direct numerical simulation (DNS) of turbulent flow over the rough bed of an open channel. We consider a hexagonal arrangement of spheres on the channel bed. The depth of flow has been taken as four times the diameter of the spheres and the Reynolds number has been chosen so that the roughness Reynolds number is greater than 70, thus ensuring a fully rough flow. A parallel code based on finite difference, domain decomposition, and multigrid methods has been used for the DNS. Computed results are compared with available experimental data. We report the first- and second-order statistics, variation of lift/drag and exchange coefficients. Good agreement with experimental results is seen for the mean velocity, turbulence intensities, and Reynolds stress. Further, the DNS results provide accurate quantitative statistics for rough bed flow. Detailed analysis of the DNS data confirms the streaky nature of the flow near the effective bed and the existence of a hierarchy of vortices aligned with the streamwise direction, and supports the wall similarity hypothesis. The computed exchange coefficients indicate a large degree of mixing between the fluid trapped below the midplane of the roughness elements and that above it.     

Turbulence Intensities of Shallow Rain-Impacted Flow over Rough Bed

A thorough understanding of the effect of raindrop impacts on the characteristics of a shallow flow is very helpful to properly predict and control water erosion. The turbulence properties of a shallow rain-impacted flow over a rough bed were studied using a two-dimensional fiber-optic laser Doppler velocimeter and an oscillating type raindrop generator. It was found that the value of momentum correction factor β for a rough boundary was much smaller than that for a smooth boundary at a given value of Reynolds number because of the vertical mixing effect of the boundary roughness. For the condition without rainfall, the vertical distribution of the longitudinal turbulence intensity became more uniform as the boundary roughness increased. For the “fully rough” condition with rainfall and constant roughness size, the effect of roughness on the distributions of turbulence intensities was small. Similar to the condition with smooth boundary, the rainfall-generated turbulence was greatest near the water surface and decreased downward at an exponential rate for the flow over rough bed. Two types of statistical correlations were developed to predict the distributions of both the longitudinal and the vertical turbulence intensities depending on whether the flow is completely penetrated by the raindrops.     

Turbulence Characteristics of Open-Channel Flow with Bed Suction

The influence of bed suction on the characteristics of turbulent open channel flow is studied in a laboratory flume using a two-component laser Doppler velocimeter. The experimental results show how bed suction significantly affects the mean flow properties, turbulence levels, and Reynolds stress distributions. The data reveal the presence of a more negative vertical (downward) velocity. The results also show how the horizontal and vertical turbulence intensities and Reynolds shear stresses respond to suction. All these properties are found to reduce with increasing relative suctions: decreasing more rapidly around the bed region than that near the free surface. In the downstream direction, the flow structure in the suction zone undergoes a process of rapid readjustment within a transitional region. Beyond this region, the turbulence flow structures asymptotes toward an “equilibrium” region.     

Velocity Distribution of Turbulent Open-Channel Flow with Bed Suction

This study investigates theoretically and experimentally the velocity distributions of turbulent open channel flow with bed suction. A velocity profile with a slip velocity at the bed surface and an origin displacement under the bed surface is proposed and discussed. Based on this assumption, a modified logarithmic law is derived. The measured experimental velocity distribution verifies the accuracy of the theoretically derived profile. The data show a significant increase in the near bed velocity and a velocity reduction near the water surface, resulting in the formation of a more uniform velocity distribution. The values of the origin displacement, slip velocity and shear velocity are found to increase with increasing relative suction. The measured data show the occurrence of two flow regions in the suction zone: a transitional region in which the velocity readjusts rapidly; and an “equilibrium” region.     

Reynolds Stress and Bed Shear in Nonuniform Unsteady Open-Channel Flow

Expressions for the Reynolds stress and bed shear stress are developed for nonuniform unsteady flow in open channels with streamwise sloping beds, assuming universal (logarithmic) velocity distribution law and using the Reynolds and continuity equations of two-dimensional open-channel flow. The computed Reynolds stress distributions are in agreement with experimental data.     

Multilayer Averaged and Moment Equations for One-Dimensional Open-Channel Flows

A model is developed to account for the vertical distribution of velocity and nonhydrostatic pressure in one-dimensional open-channel flows. The model is based on both classical multilayer models and depth-averaged and moment equations. The establishment of its governing equations and the flow simulation are performed over a number of flow layers as in classical multilayer models. However, the model also allows for vertical distributions within a flow layer by including both Boussinesq terms and effective stress terms due to depth-averaging operations. These terms are evaluated on the basis of vertically linearly approximated profiles of velocity and pressure. The resulting additional coefficients can be solved by the moment equations for the relevant layers. Three verifications demonstrate satisfactory simulations for water surface profile, as well as vertical distributions for horizontal velocity, vertical velocity, and nonhydrostatic pressure. Sensitivity analysis shows that the model can be applied with fewer flow layers, more flexibility of layer division, and less computational cost than classical multilayer models, without a remarkable compromise in accuracy.     

Flow in Open-Channel Embayments

This study investigates flows in a square and a rectangular embayment located on the side bank of an open channel. It is found that the main flow of the open channel induces a circulatory flow in the square and rectangular embayment and the center of the circulatory flow is shifted downstream in comparison with the center of the embayment. A solution of the shallow water equations solved using the method of variation yields results of the 95% confidence intervals within 10% of mean errors between the observed and computed nondimensional velocities.     

Response of Velocity and Turbulence to Sudden Change of Bed Roughness in Open-Channel Flow

The experimental study shows how an open-channel flow would respond to a sudden change (from smooth to rough) in bed roughness. Using a two-dimensional acoustic Doppler velocimeter and a laser Doppler velocimeter, the velocity, turbulent intensities, and Reynolds stress profiles at different locations along a laboratory flume were measured. Additionally, the water surface profile was also measured using a capacitance-type wave height meter. The experimental data show the formation of an internal boundary layer as a result of the step change in bed roughness. The data show that this boundary layer grows much more rapidly than that formed in close-conduit flows. The results also show that the equivalent bed roughness, bed-shear stress, turbulent intensities, and Reynolds stress change gradually over a transitional region, although the bed roughness changes abruptly. The behavior is different from that observed in close-conduit flows, where an overshooting property—which describes the ability of the bed-shear stress to attain a high-peak value over the section with the larger roughness, was reported. A possible reason for the difference is the variation of the water surface profile when an open-channel flow is subjected to a sudden change in bed roughness.     

Frequency Modeling of Open-Channel Flow

A good model is necessary to design automatic controllers for water distribution in an open-channel system. The frequency response of a canal governed by the Saint-Venant equations can be easily obtained in the uniform case. However, in realistic situations, open-channel systems are usually far from the uniform regime. This paper provides a new computational method to obtain a frequency domain model of the Saint-Venant equations linearized around any stationary regime (including backwater curves). The method computes the frequency response of the Saint-Venant transfer matrix, which can be used to design controllers with classical automatic control techniques. The precision and numerical efficiency of the proposed method compare favorably with classical numerical schemes (e.g., Runge–Kutta). The model is compared in nonuniform situations to the one given by a finite difference scheme applied to Saint-Venant equations (Preissmann scheme), first in the frequency domain, then in the time domain. The proposed scheme can be used, e.g., to validate finite difference schemes in the frequency domain.     

Reynolds Stress Anisotropy in Open-Channel Flow

This paper reports the results of an experimental study characterizing turbulence and turbulence anisotropy in smooth and rough shallow open-channel flows. The rough bed consists of a train of two-dimensional transverse square ribs with a ratio of the roughness height (k) to the total depth of flow (d) equal to 0.10. Three rib separations (p/k) of 4.5, 9, and 18 were examined. Here, p is the pitch between consecutive roughness elements and was varied to reproduce the classical condition of d- and k-type roughness. For each case, two-component velocity measurements were obtained using a laser Doppler velocimetry system at two locations for p/k = 4.5 and 9: on the top of the rib and above the cavity, and an additional location for p/k = 18. The measurements allow examination of the local variations of the higher-order turbulent moments, stress ratios as well as turbulence anisotropy. Large variations of the turbulence intensities, Reynolds shear stress, turbulent kinetic energy and turbulence production are found for y1<3k. In this region, the flow is more directly influenced by the shear layers from the preceding ribs. The higher-order moments appear to be similar for all rough surfaces beyond y1 ≈ 7k. In the outer layer (y1>3k), all higher-order turbulent moments for the k-type roughness show a substantial increase due to the complex interactions between the roughness and the remnants overlying shear layers shed from succeeding ribs. Analysis of the components of the Reynolds stress anisotropy tensor shows that at p/k = 18, the flow at y1<5k tends to be more isotropic which implies that for this particular case, the effect of the roughness density could also be important. On the smooth bed, at the shallower depths, the correlation coefficient near the free surface increases and turbulence tends to become less anisotropic.     

Wall Similarity in Turbulent Open-Channel Flow

The present work demonstrates that the assumption of wall similarity in the equilibrium region of open channels leads to the definition of a constant, universal value for the normalized vertical flux of turbulent kinetic energy irrespective of wall roughness. Moreover, by additionally assuming a gradient-flux model for the turbulent kinetic energy, its bed-normal distribution is easily estimated by an expression with parameters that rely on a physical basis. Turbulence measurements performed in smooth-, transitionally rough-, and fully rough-bed (sandgrain) open channels confirm this hypothesis, providing at the same time an alternative way of determining the lower and upper limits of the intermediate region. Finally, based upon wall-similarity assumptions, a new methodology is proposed for estimating mean bed-shear velocities (stresses) in free-surface flows at high local and global Reynolds numbers, irrespective of roughness condition at the bed.     

Parameter Estimation for Flow in Open-Channel Networks

Two mathematical models for parameter estimation in closed-loop, open-channel flow networks are presented. The parameter estimation models seek to determine the parameter values that would reproduce an observed flow profile in the channel network. The governing equations for gradually varied flow in channel networks are the optimization models’ constraints. The projected augmented Lagrangian method is used to solve the optimization models. Performance of these two optimization models is evaluated for a given closed-loop network configuration. Results establish the potential of the developed models for use in real-life flow scenarios.     

Maximum Velocity and Regularities in Open-Channel Flow

Maximum velocity in a channel section often occurs below the water surface. Its location is linked to the ratio of the mean and maximum velocities, velocity distribution parameter, location of mean velocity, energy and momentum coefficients, and probability density function underpinning a velocity distribution equation derived by applying the probability and entropy concepts. The mean value of the ratio of the mean and maximum velocities at a given channel section is stable and constant, and invariant with time and discharge. Its relationship with the others in turn leads to formation of a network of related constants that represent regularities in open-channel flows and can be used to ease discharge measurements and other tasks in hydraulic engineering. Under the probability concept, the ratio of mean and maximum velocities being constant means that the probability distribution underpinning the velocity distribution and other related variables is resilient, and that the same probability distribution is governing various phenomena observable at a channel section and explains the regularities in open-channel flows.     

Subcritical 90° Equal-Width Open-Channel Dividing Flow

Based on experimental observations, for a subcritical, right-angled, equal-width, open-channel dividing flow over a horizontal bed, the contraction coefficient at the maximum width-contracted section in the recirculation region is almost inversely related to the main channel upstream-to-downstream discharge ratio. The energy heads upstream and downstream of the division in the main channel are found to be almost equal. Under the assumption that the velocities are nearly uniformly distributed at the considered boundaries, the depth-discharge relationship follows the commonly used energy equation. The predicted results correlate fairly with the experimental data from this and other studies. The energy-loss coefficient of a division is expressed in terms of discharge ratio, upstream Froude number, and depth ratio. An expression for practical engineering applications is to determine the maximum possible branch-channel discharge at a given upstream discharge with a prescribed downstream Froude number or the maximum possible downstream Froude number if both branch- and main-channel discharges are prescribed.     

Modeling Unsteady Open-Channel Flow for Controller Design

An approximated linear model of unsteady open-channel flow is necessary to design the water-level controller for irrigation open channels. Toward this end, this paper presents the matrix approach to derive the linear model of open-channel system in analytical form mainly according to the Saint Venant equations and the backwater profile at the steady state of open channel. The hydraulic model of the check structure at the downstream end of open channel is also incorporated into the linear model. A practical example indicates that the frequency response of the open-channel system can be accurately analyzed with the linear model. The simulation results in the time domain show that the dynamic behavior of the linear model approximates to that of the nonlinear model of the open-channel system. Finally, the limitations of the linear model are discussed.     

Critical Depth Relationships in Developing Open-Channel Flow

Doubts have been expressed about the validity of the critical depth defined in terms of the minimum specific energy head of the free-surface streamline when dealing with developing open-channel flows. This note examines the two approaches for defining critical flow, that based on the minimum specific energy of the free-surface streamline and that based on the mean energy head of the whole flow section. Large differences for the dimensionless critical depths are obtained with the two methods due to each critical depth proving to be a different control point on the free-surface profile. It is argued that both methods are different alternatives, although the critical depth concept was different in each case. Theoretical support to critical flow computations based on the free streamline is provided. An alternative approach for computing the discharge characteristics of broad-crested weirs based on the energy loss inside the boundary layer is also proposed.     

Mean Flow and Turbulence in Open-Channel Bend

Flow over a developed bottom topography in a bend has been investigated experimentally. The measuring section is in the outer-bank half of the cross section at 60° into the bend. Spatial distributions of the mean velocities, turbulent stresses, and mean-flow and turbulent kinetic energy are presented. The cross-sectional motion contains two cells of circulation: besides the classical helical motion (center-region cell), a weaker counterrotating cell (outer-bank cell) is observed in the corner formed by the outer bank and the water surface. The downstream velocity in the outer half-section is higher than the one in straight uniform flow; the core of maximum velocities is found close to the separation between both circulation cells, well below the water surface. The turbulence structure in a bend is different from that in a straight flow, most notably in a reduction of the turbulent activity toward the outer bank. Both the outer-bank cell and reduced turbulent activity have a protective effect on the outer bank and the adjacent bottom and thus influence the stability of the flow perimeter and the bend morphology.