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.     

Velocity Distribution in the Roughness Layer of Rough-Bed Flows

Several models for the vertical distribution of the double-averaged (in time and in the plane parallel to the mean bed) longitudinal velocity in the flow region between roughness troughs and roughness tops are suggested. We found that the same model for velocity distribution may be applicable to a range of flow conditions and roughness types, which share some common features. The suggested models for velocity distribution in the near-bed region are: (1) Constant velocity; (2) exponential velocity distribution; and (3) linear velocity distribution. The measured velocity distributions may be approximated by a single model or by a combination of models depending on roughness geometry and flow conditions. The validity of these models for velocity distribution is supported by laboratory data.     

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.     

Depth-Averaged Shear Stress and Velocity in Open-Channel Flows

Turbulent momentum and velocity always have the greatest gradient along wall-normal direction in straight channel flows; this has led to the hypothesis that surplus energy within any control volume in a three-dimensional flow will be transferred toward its nearest boundary to dissipate. Starting from this, the boundary shear stress, the Reynolds shear stress, and the velocity profiles along normal lines of smooth boundary may be determined. This paper is a continuous effort to investigate depth-average shear stress and velocity in rough channels. Equations of the depth-averaged shear stress in typical open channels have been derived based on a theoretical relation between the depth-averaged shear stress and boundary shear stress. Equation of depth mean velocity in a rough channel is also obtained and the effects of water surface (or dip phenomenon) and roughness are included. Experimental data available in the literature have been used for verification that shows that the model reasonably agrees with the measured data.     

Analytic Stage-Discharge Formulas for Flow in Straight Prismatic Channels

Three analytic stage-discharge formulas, suitable for hand calculation, have been derived for prismatic open channels. The three types of prismatic geometry investigated are: simple rectangular, symmetric, and asymmetric rectangular compound channels. The formulas include three key parameters that govern the local friction factor, eddy viscosity, and secondary flow. The discharge results and the corresponding analytic depth-averaged velocity and bed shear stress distributions show good agreement with the experimental data, even when constant parameter values are assumed. The influence of these three parameters on the stage-discharge relationships is explored.     

Flow through Trapezoidal and Rectangular Channels with Rigid Cylinders

A physical model study was performed in which wood dowels were used to model rigid vegetation. The dowel configurations used in the flume were intended to simulate the effects of willow post systems (i.e., collections of rigid cylinders placed along a streambank to reduce streambank erosion). In addition, an analytical model is presented for predicting depth-averaged velocity distributions in straight trapezoidal or rectangular channels with newly constructed willow post systems. The analytical model is founded on wake theory and is applicable to channels with submerged and unsubmerged rigid cylinders. Data from three independent physical model studies were used to validate the analytical model. Depth-averaged velocities predicted using the analytical model, Upred, were compared to velocities observed in the physical models, Uobsd, and yielded discrepancy ratios, Upred/Uobsd, that were typically between 0.80 and 1.20. Results from this study are that significant variables for reducing local velocities are cylinder height, diameter, and density (number of cylinders per unit area); and the arrangement of the cylinders (i.e., rectangular versus staggered grid) is inconsequential.     

End Depth–Discharge Relation at Free Overfall of Trapezoidal Channels

For steady flow near the free overfall (end section) of a horizontal trapezoidal channel, the velocity distribution is nonuniform and the streamlines are curved. An accurate relation between discharge rate and end depth was formulated including these effects. To determine these effects, the streamline pattern in the vertical plane of channel symmetry was determined using measured velocity components and the water surface profile. At the end section, the streamline pattern yielded the streamline curvature, which in turn provided the curvature correction required to predict the true static pressure head profile. The measured static pressure head distribution agreed well with the predicted static pressure head distribution for the end section. The pressure force at the end section was obtained on the basis of the measured static pressure distribution at the end section, and this information yielded a reliable relation between the end depth and the channel discharge rate. Analysis of present and past experimental data indicated that, the pressure head coefficient was the dominant parameter that influences the relationship between discharge rate and end depth in trapezoidal channels. Near the end depth, in the region above the maximum velocity point, the total energy determined from the measured velocity and pressure fields was essentially constant.     

Boundary Shear Distribution in Straight Ducts and Open Channels

A semianalytical model was developed to predict boundary shear distribution in straight, noncircular ducts and open channels. The model was developed using a simplified streamwise vorticity equation, which involves only secondary Reynolds stress terms. These terms are representative of transverse turbulence anisotropy and nonhomogeneity. Transverse anisotropy is modeled using a universal function. Shear stresses are incorporated into the model by applying the momentum transfer model. An empirical model was employed to calculate the effect of the channel boundary on shear stresses. The final equation was applied to calculate boundary shear distribution in triangular ducts and trapezoidal open channels. The model predictions were well correlated with experimental data.     

Lateral Depth-Averaged Velocity Distributions and Bed Shear in Rectangular Compound Channels

This paper presents a method to predict depth-averaged velocity and bed shear stress for overbank flows in straight rectangular two-stage channels. An analytical solution to the depth-integrated Navier–Stokes equation is presented that includes the effects of bed friction, lateral turbulence, and secondary flows. The Shiono and Knight method accounts for bed shear, lateral shear, and secondary flow effects via three coefficients, f, λ, and Γ, respectively. A novel boundary condition at the internal wall between the main channel and the adjoined floodplain is proposed and discussed along with other conventional boundary conditions. The analytical solution using the novel boundary condition gives good prediction of both lateral velocity distribution and bed shear stress when compared with experimental data for different aspect ratios.     

Flow and Velocity Distribution in Meandering Compound Channels

An investigation of flow and velocity distribution in meandering compound channels with over bank flow is described. Equations concerning the three-dimensional variation of longitudinal, transverse, and vertical velocity in the main channel and floodplain of compound section in terms of channel parameters are presented. The flow and velocity distributions in meandering compound channels are strongly governed by interaction between flow in the main channel and that in the floodplain. The proposed equations take adequate care of the interaction affect. Results from the formulations, simulating the three-dimensional velocity field in the main channel and in the floodplain of meandering compound channels are compared with their respective experimental channel data obtained from a series of symmetrical and unsymmetrical test channels with smooth and rough sections. The aspect ratio of the test channels varies from two to five. The equations are found to be in good agreement with the experimental data. The formulations are verified against the natural river and other meandering compound channel data. The power laws used for simulating the three-dimensional velocity structure are found to be quite adequate.     

Depth-Averaged Two-Dimensional Numerical Modeling of Unsteady Flow and Nonuniform Sediment Transport in Open Channels

A depth-averaged two-dimensional (2D) numerical model for unsteady flow and nonuniform sediment transport in open channels is established using the finite volume method on a nonstaggered, curvilinear grid. The 2D shallow water equations are solved by the SIMPLE(C) algorithms with the Rhie and Chow’s momentum interpolation technique. The proposed sediment transport model adopts a nonequilibrium approach for nonuniform total-load sediment transport. The bed load and suspended load are calculated separately or jointly according to sediment transport mode. The sediment transport capacity is determined by four formulas which are capable of accounting for the hiding and exposure effects among different size classes. An empirical formula is proposed to consider the effects of the gravity on the sediment transport capacity and the bed-load movement direction in channels with steep slopes. Flow and sediment transport are simulated in a decoupled manner, but the sediment module adopts a coupling procedure for the computations of sediment transport, bed change, and bed material sorting. The model has been tested against several experimental and field cases, showing good agreement between the simulated results and measured data.     

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.     

Multilayer Depth-Averaged Flow Model with Implicit Interfaces

Using a depth-averaged model to obtain the velocity and pressure distributions in the vertical direction is difficult. A multilayer model is an option that can be used to improve on the depth-averaged model. However, the unknown flow depth needs to be predicted first and then divided into layers as an input for the multilayer model. An improved multilayer model is proposed here by introducing an implicit layer dividing interfaces that are associated with the flow velocity and pressure distribution. The formulation of interfaces also applies to boundary faces: Free surface and channel beds. Therefore, each flow layer behaves like that in the classical depth-averaged model. Subsequently the governing equations are also simplified due to the vanishing terms related to interfacial flow exchanges. This improved model has been satisfactorily applied to steady flow simulations in three cases: Flow over a slope transition from mild to steep, from steep to mild, and over a trapezoidal weir. The results demonstrate the efficiency and validity of the proposed models to simulate open channel flows with bed slope changes.     

Velocity Distribution and Energy Dissipation along Stepped Chutes Lined with Wedge-Shaped Concrete Blocks

Stepped channels lined with wedge-shaped concrete blocks may constitute a low-cost alternative to provide overtopping protection of embankment dams if the discharge capacity of existing spillways is not adequate or even to be used as the main spillway of newly built embankment dams. This paper addresses the velocity distribution and the energy dissipation, downstream of the inception point, on stepped chutes lined with wedge-shaped concrete blocks. An experimental setup was developed with two flumes designed with a relative scale of 1∶2.5. Air concentration was measured with an optical probe in several cross sections of both flumes. The velocity profiles along chutes lined with wedge-shaped blocks with the upper face sloping downstream were analyzed. The measurements’ accuracy was checked by comparing discharges indicated by a facility flowmeter and obtained by the integration of velocity and air concentration profiles. The effect of the steps-slope in the energy dissipation is studied. Values of the Darcy-Weissbach friction factor are proposed for this type of chute lining, for transition flows, and for skimming flows.     

Side Outflow from Supercritical Channel Flow

Side flow on a flood plain from a side outlet of the main channel is investigated both theoretically and experimentally for supercritical main flow. The side outlet as a model simulates a failure of a river bank in a prototype. The discharge ratios of the side outflow to that of the main channel flow, the water depth, and the velocity at the side outlet are obtained. The theoretical discharge ratio is a function of the Froude number of the main channel flow. The theoretical spreading angle of the side flow and the theoretical relationship between the velocities and the distance from an upstream point of the side outlet are also compared and predicted. All the theoretical results are found to be in good agreement with the experimental data.     

Unifying Criterion for the Velocity Reversal Hypothesis in Gravel-Bed Rivers

It has been hypothesized that velocity reversals provide a mechanism for maintaining pool-riffle morphology in gravel-bed rivers—an important habitat for salmonids, which are at risk in many places worldwide and that are the focus of extensive environmental legislation in Europe and North America. However, the occurrence of velocity reversals has been controversial for over 3 decades. We present a simple one-dimensional criterion that unifies and explains previous disparate findings regarding the occurrence of velocity reversals. Results show that reversal depends critically on the ratio of riffle-to-pool width, residual pool depth (difference between pool and riffle elevations), and on the depth of flow over the riffle, suggesting that land management activities which alter channel form or divert water from the channel can have negative impacts on the sustainability of pool-riffle habitat in gravel-bed rivers.     

Mathematical Modeling of Meandering Channels with a Generalized Depth Averaged Model

A two-dimensional depth averaged model is developed in a nonorthogonal curvilinear coordinate system. The ability of the model in handling complex mesh arrangements with high aspect ratios and skewness is investigated. It is shown that model formulation makes it able to handle large skewness in the grid lines. It is also shown that in predicting the water surface profile with a very distorted mesh, most of the errors arise from the large aspect ratio rather than the skewness of the grid lines. The model is then applied to three meandering channels (two simple and one compound), with specifications similar to those found in nature and the results are compared with experimental data. The comparison shows that the model predicted the water surface profile and velocity distribution well in simple channels. Predictions of the model in the main channel of the compound meandering channel were also in general agreement with the experimental results.     

Overbank Flow in Compound Channels with Nonprismatic Floodplains

In this paper, the authors present experimental results of overbank flow in compound channels with nonprismatic floodplains and different convergence angles. The depth-averaged velocity, the local velocity distributions, and the boundary shear stress distributions were measured along the converging flume portion for different relative depths. The momentum balance is used to analyze the force acting on the flow in the main channel and for the whole cross section. Using the experimental data, various terms in the momentum equation are also calculated. The apparent shear forces on the vertical interface between the main channel and floodplains are evaluated for compound channels with nonprismatic floodplains, and the results are then compared with the prismatic cases. The energy balance in nonprismatic compound channels is also investigated by using the water surface elevation.