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.     

Turbulence Characteristics in Gradual Channel Transition

A field study was conducted to determine the effects of a channel transition on turbulence characteristics. Detailed three-dimensional (3D) flow measurements were collected at a cross section that is located downstream of a gradual channel expansion. These measurements were obtained via an acoustic doppler velocimeter and include the 3D velocity field, the mean local velocities, the turbulent intensities, the frictional characteristics of the flow, the secondary velocity along the transverse plane, and the instantaneous shear stress components in the streamwise and transverse directions. Analysis of the 3D flow data indicates that the turbulent flow on the outer bank of the channel is anisotropic. Such anisotropy of turbulence, which is attributed to the gradual expansion in the channel and bed roughness, yields the development of a secondary flow of Prandtl’s second kind as reported in 1952. In particular, it was found that turbulent intensities in the vertical and transverse directions on the outer bank section are different in magnitude creating turbulence anisotropy in the cross-sectional plane and secondary flows of the second kind. Turbulent intensities increase toward the free surface indicating the transfer of a higher-momentum flux from the channel bed to the free surface, which contradicts common wisdom. Results for the normalized stress components in the streamwise and transverse direction show similar behavior to the intensities. Moreover, the nonlinear distribution of stresses is indicative of the oscillatory nature of the flow induced by the secondary flows of Prandtl’s second kind. A similar behavior was found for flows in straight rectangular channels over different roughness. Finally, a comparison between the secondary current velocity with the mainstream velocity indicates that secondary flow of Prandtl’s second kind is present within the right half of the measured cross section.     

Distribution of Bed Shear Stress in Rotating Circular Flume

Distributions of bed shear stress across the width of a rotating circular flume with smooth and rough bed surfaces were obtained by measurement and model prediction. Results with flows over smooth beds showed that the flow in the central part may be considered to be two-dimensional and that effects of flow depth over the operating range of the flume are minor for flow depths not exceeding 0.14 m. For rough beds, the bed shear stress distributions were found to be skewed toward the inner wall. This can be corrected if a compensating roughness is added to the bottom of the ring. Such measures are also effective for flumes with smooth beds. Measured bed shear stress distributions agreed well with the predicted distributions for smooth beds and reasonably well for rough beds. The modified Preston tube, for measurement of bed shear stress in flows over rough beds, was found to give promising results. Further tests are required to completely define the uncertainty in bed shear stress measurements made with this instrument.     

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.     

Fluctuations of Turbulent Bed Shear Stress

With the assumption that the bed shear stress fluctuates in a lognormal fashion, the probability density function (PDF) of the standardized bed shear stress is derived as a function of the relative shear stress intensity. The PDF is more skewed with larger relative intensities, but approaches a Gaussian function when the relative intensity is small. The computed PDF agrees well with the reported experimental data for flows over a smooth boundary. The higher-order moments of the bed shear stress, skewness, and kurtosis, are shown analytically to be also dependent on the relative intensity. The theoretical dependencies are then compared to a number of measurements available in the literature. The Reynolds number effect on the relative intensity is also discussed.     

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.     

Characteristics of Shear Layer Structure in Skimming Flow over a Vertical Drop Pool

The characteristics of shear layer structure between the sliding jet and the pool for skimming flows over a vertical drop pool were investigated experimentally, using flow visualization technique and high speed particle image velocimetry. Four series of experiments having different end sill ratios (h/H = 0.12, 0.43, 0.71 and 1.0, where h=end sill height and H=drop height) with various approaching flow discharges were performed to measure the detailed quantitative velocity fields of the shear layer. The mean velocities and turbulence properties were obtained by ensemble averaging the repeated measurements. From the velocity profiles, it is found that the growth of the shear layer in the downward direction as the jet slides down the pool represents the momentum exchange. Analyzing the distribution of measured velocity, the similarity profile of the mean velocity at different cross sections along the shear layer was obtained. The proposed characteristic scales provided unique similarity profiles having promising regression coefficient. The selection of these characteristic scales is also discussed. Further, the spatial variations of mean velocity profiles, turbulence intensities, in-plane turbulent kinetic energy, and Reynolds shear stress were also elucidated in detail. The imperative observation is that the Reynolds shear stress dominates the major part along the shear layer as compared to the viscous shear stress. The study also provides an insight into the flow phenomena through the velocity and turbulent characteristics.     

Wall-Wake Flows Downstream of a Sphere Placed on a Plane Rough Wall

The time-averaged characteristics of turbulent wall-wake flows downstream of a sphere placed on a rough wall are studied. The profiles of the defect of streamwise velocity, Reynolds shear stress, and turbulence intensities exhibit some degree of similarities when they are scaled by their respective peak defect values. For the velocity defect profiles, the vertical distances are scaled by the height of the location of the half-peak velocity defect. However, for the defect profiles of the Reynolds shear stress and the turbulence intensities, the vertical distances are scaled by the height of the location of the half-peak Reynolds shear stress defect. The magnitudes of the peak defect of all the quantities diminish with the distance downstream of the sphere characterizing the recovery of their undisturbed profiles. Additionally, the theoretical similarity solution for the velocity defect profiles is obtained. The third-order correlations imply that in the inner layer of wall wakes, a streamwise acceleration is prevalent and associated with a downward flux, suggesting sweeps. In contrast, in the outer layer, a streamwise deceleration exists and is associated with an upward flux, suggesting ejections. The profiles of the energy budget show that the turbulent and pressure energy diffusions oppose each other. The turbulent production has a positive peak, and the pressure energy diffusion has a negative peak, indicating a large gain in turbulence production in the wall-wake flows. The quadrant analysis confirms that in wall-wake flows, sweeps are the governing mechanism resulting from an inrush of fluid streaks. The bursting events have shorter duration, but they are more frequent than those in upstream.     

Numerical Modeling of Three-Dimensional Flow Field Around Circular Piers

A three-dimensional numerical model FLUENT is used to simulate the separated turbulent flow around vertical circular piers in clear water. Computations are performed using different turbulence models and results are compared with several sets of experimental data available in the literature. Despite commonly perceived weakness of the k-ε model in resolving three-dimensional (3D) open channel and geophysical flows, several variants of this turbulence model are found to have performed satisfactorily in reproducing the measured velocity profiles. However, model results obtained using the k-ε models show some discrepancy with the measured bed shear stress. The Reynolds stress model performed quite well in simulating velocity distribution on flat bed and scour hole as well as shear stress distribution on flat bed around circular piers. The study demonstrates that a robust 3D hydrodynamic model can effectively supplement experimental studies in understanding the complex flow field and the scour initiation process around piers of various size, shape, and dimension.     

Formula for Sediment Transport in Rivers, Estuaries, and Coastal Waters

The aim of the present study is to develop a formula for the relationship between flow strength and sediment discharge. The appropriate definition of energy dissipation rate E in the theorem of Bagnold in 1966 is discussed and it is found that the sediment transport rate gt in unidirectional flows can be well predicted when E is defined as the product of bed shear stress τ0 and near bed velocity u*′. Then the linear relationship between u*′E and the sediment transport rate is examined using measured data. The good agreement between measured and predicted values indicates that the phenomena of sediment transport can be reasonably described by the near bed flow characteristics. As the hydrodynamic modelers are able to calculate the bed shear stress and near bed velocity in various cases now, thus the new relationship may provide numerical modelers a tool to calculate the sediment transport in rivers, estuaries and coastal waters. To prove this, the simplified analytical expressions of E and u*′ in wave-current flows and coastal waters are derived, the results are checked with the available data over a wide range of flow conditions; and good agreements are achieved, indicating that the presumption is valid in the cases investigated.     

Near-Bed Turbulence Characteristics at the Entrainment Threshold of Sediment Beds

This experimental study is devoted to quantification of the near-bed turbulence characteristics at an entrainment threshold of noncohesive sediments. Near the bed, the departure in the distributions of the observed time-averaged streamwise velocity from the logarithmic law is more for immobile beds than for entrainment-threshold beds. In the Reynolds shear stress distributions, a damping that occurs near the bed for sediment entrainment is higher than that for immobile beds. Quadrant analysis reveals that in the near-bed flow zone, ejections and sweeps on immobile beds cancel each other, giving rise to the outward interactions, whereas sweeps are the dominant mechanism toward sediment entrainment. The bursting duration for entrainment-threshold beds is smaller than that for immobile beds. On the other hand, the bursting frequency for entrainment-threshold beds is larger than that for immobile beds. The third-order correlations indicate that during sediment entrainment, a streamwise acceleration associated with a downward flux and advection of streamwise Reynolds normal stress is prevalent. The streamwise and the downward vertical fluxes of turbulent kinetic energy (TKE) increase with sediment entrainment. The TKE budget proves that for sediment entrainment, the pressure energy diffusion changes drastically to a negative magnitude, indicating a gain in turbulence production.     

Probability Density Function of Instantaneous Drag Forces and Shear Stresses on a Bed

A probability density function (PDF) of the instantaneous bed shear stress in a turbulent flow is derived in this paper. It is argued that the shape of the PDF is similar to the PDF of the instantaneous drag forces on bed roughness elements. The influence of the near-bed relative turbulence intensity is included in the PDF. The shape of the distribution compares well with our measurements of the instantaneous drag force on a protruding bed element for a range of turbulence intensities. However, deviations are apparent at high turbulence intensities. The PDF also compares well with measurements of shear stresses on a smooth wall.     

Turbulence and Secondary Flow over Sediment Stripes in Weakly Bimodal Bed Material

Longitudinal stripes are a common bed form in heterogeneous alluvial sediments and consist of periodic, spanwise variations in bed texture and elevation that are aligned parallel to the mean flow direction. This paper quantifies mean and turbulent flow structures over self-formed sediment stripes in a weakly bimodal sand and gravel mixture. Turbulence anisotropy generates two secondary circulation cells across the channel half-width, which produce a cross-stream perturbation in boundary shear stress. The interaction between this flow structure and the selective transport of bed material generates spanwise sediment sorting that is symmetrical about the centerline. Finer sediments are entrained from regions of high shear stress, transported laterally by the secondary flow, and deposited in regions of lower shear stress. Lateral changes in bed texture further enhance the near-bed secondary flow, which provides a positive feedback mechanism for stripe growth. In bimodal sediments, at shear stresses just above the entrainment threshold, stripes may replace lower-stage plane beds. At higher shear stresses the coarser sediment becomes more mobile and the stripes are replaced by flow transverse bed forms.     

Three-Dimensional Numerical Simulation of Compound Channel Flows

A three-dimensional numerical study is presented for the calculation of turbulent flow in compound channels. The flow simulations are performed by solving the three-dimensional Reynolds-averaged continuity and Navier–Stokes equations with the k?ε turbulence model for steady-state flow. The flow equations are solved numerically with a general-purpose finite-volume code. The results are compared with the experimental data obtained from the UK Flood Channel Facility. The simulated distributions of primary velocity, bed shear stress, turbulent kinetic energy, and Reynolds stresses are used to investigate the accuracy of the model prediction. The results show that, using an estimated roughness height, the primary velocity distributions and the bed shear stress are predicted reasonably well for inbank flows in channels of high aspect ratio (width/depth ≥ 10) and for high overbank flows with values of the relative flow depth greater than 0.25.     

Characteristics of Turbulent Flow in Submerged Jumps on Rough Beds

The results of an experimental investigation on the flow field in submerged jumps on horizontal rough beds, detected by an acoustic Doppler velocimeter, are presented. Experiments were conducted for the conditions of submerged jumps, having submergence factors from 0.96 to 1.85 and jet Froude numbers from 2.58 to 4.87, over rough beds of Nikuradse’s equivalent sand roughness equaling 0.49, 0.8, 1.86, and 3?mm. The vertical distributions of time-averaged velocity components, turbulence intensity components, and Reynolds stress at different streamwise distances from the sluice opening and the horizontal distribution of bed-shear stress are plotted. Vector plots of the flow field show that the rate of decay of jet velocity in a submerged jump increases with increase in bed roughness. The flow characteristics on rough beds, being different from those on smooth bed, are discussed from the point of view of similarity, growth of the length scale, and decay of the velocity and turbulence characteristics scales. The most important observation is that the flow in the fully developed zone is found to be self-preserving.     

Double-Averaged Open-Channel Flows with Small Relative Submergence

We investigate the turbulent structure of shallow open channel flows where the flow depth is too small (compared with the roughness height) to form a logarithmic layer but large enough to develop an outer layer where the flow is not directly influenced by the roughness elements. Since the log layer is not present, the displacement height d, which defines the position of the zero plane, and the shear velocity u* cannot be found by fitting the velocity data to the log law. However, these parameters are still very important because they are used for scaling flow statistics for the outer and roughness layers. In this paper we propose an alternative procedure for evaluating d in laboratory conditions, where d is found from additional experiments with the fully developed log layer. We also point out the appropriate procedure for evaluating the shear velocity u* for flows with low submergence. These procedures are applied to our own laboratory flume experiments with uniform sphere roughness, where velocities were measured using Particle Image Velocimetry. Results were interpreted within the framework of the double-averaged Navier–Stokes equations and include mean velocities, turbulence intensities, Reynolds stresses, and form-induced normal and shear stresses. The data collapse well and show that in flows without a developed log layer the structure of turbulence in the outer layer remains similar to that of flows with a log layer. This means that even though the roughness layer in the experiments reported herein was sufficiently high to prevent the development of the log layer, influence of the bed roughness did not spread further up into the outer layer. Furthermore, the results show that flow statistics do not depend on relative submergence except for the form-induced stresses which increase when relative submergence decreases.     

Effect of Upward Seepage on Scour and Flow Downstream of an Apron due to Submerged Jets

The upward seepage through the bed sediment downstream of an apron of a sluice gate structure is a common occurrence due to afflux of the flow level between the upstream and downstream reaches of a sluice gate. The result of an experimental investigation on the characteristics of the scour hole and the flow-field downstream of an apron due to submerged jets under the influence of upward seepage through the bed sediment is presented. Experiments were run for the conditions of submerged jets, having submergence factors from 0.99 to 1.72 and jet Froude numbers from 3.15 to 4.87, over beds of sediments (median sizes = 0.8, 1.86, and 3?mm) downstream of an apron under upward seepage velocities. The characteristic lengths of the scour hole determined from the scour profiles are: the maximum equilibrium scour depth, the horizontal distance of the location of maximum scour depth from the edge of the apron, the horizontal extent of the scour hole from the edge of the apron, the dune height, and the horizontal distance of the dune crest from the edge of the apron, all of which were found to increase with an increase in the seepage velocity. Using experimental results, the time variation of the scour depth is scaled by an exponential law, where the nondimensional time scale decreases linearly with an increase in the ratio of the seepage velocity to the issuing jet velocity. The flow field in the submerged jets over both the apron and within the scour hole was detected using an acoustic Doppler velocimeter. The vertical distributions of time-averaged velocities, turbulence intensities and Reynolds stress at different streamwise distances, and the horizontal distribution of bed-shear stress are plotted for the conditions of scour holes with and without upward seepage. Vector plots of the flow field show that the rate of decay of the submerged jet decreases with an increase in the seepage velocity. The flow characteristics in the scour holes are analyzed in the context of the influence of upward seepage velocity on the decay of the velocity and turbulence intensities and the growth of the boundary layer.     

Bed-Load Effects on Hydrodynamics of Rough-Bed Open-Channel Flows

The extent to which turbulent structure is affected by bed-load transport is investigated experimentally using a nonporous fixed planar bed comprising mixed-sized granular sediment with a d50 of 1.95?mm. Three different sizes of sediment (d50 = 0.77, 1.99, and 3.96?mm) were fed into the flow at two different rates (0.003 and 0.006?kg/m/s), and subsequently transported as bed load. Particle image velocimetry (PIV) was used to determine the turbulence characteristics over the fixed bed during clear water and sediment feed cases. Mean longitudinal flow velocities at any given depth were lower than their clear water counterparts for all but one of the mobile sediment cases. The exception was with the transport of fine grains at the higher feed rate. In this case, longitudinal mean flow velocities increased compared to the clear water condition. The coarse grains tended to augment bed roughness, but fine grains saturated the troughs and interstices in the bed topography, effectively causing the influence of bed irregularities to be smoothed. The PIV technique permitted examination of both temporal and spatial fluctuations in flow variables: therefore many results are presented in terms of double-averaged quantities (in temporal and spatial domains). In particular, the form-induced stress, which arises from spatially averaging the Reynolds averaged Navier–Stokes equations and is analogous to the Reynolds turbulent stress, contributed between 15 and 35% of the total measured shear stress in the roughness layer. Flow around protrusive roughness elements produced a significant proportion of the turbulent kinetic energy shear production, suggesting that this process is highly intermittent near rough beds.     

Spatially Averaged Open-Channel Flow over Rough Bed

In this paper it is suggested that the double-averaged (in temporal and in spatial domains) momentum equations should be used as a natural basis for the hydraulics of rough-bed open-channel flows, especially with small relative submergence. The relationships for the vertical distribution of the total stress for the simplest case of 2D, steady, uniform, spatially averaged flow over a rough bed with flat free surface are derived. These relationships explicitly include the form-induced stresses and form drag as components of the total stress. Using this approach, we define three types of rough-bed flows: (1) Flow with high relative submergence; (2) flow with small relative submergence; and (3) flow over a partially inundated rough bed. The relationships for the double-averaged velocity distribution and hydraulic resistance for all three flow types are derived and compared with measurements where possible. The double-averaged turbulent and form-induced intensities and stresses for the case of regular spherical-segment-type roughness show the dominant role of the double-averaged turbulence stresses and form drag in momentum transfer in the near-bed region.