Two-Phase Flow Characteristics of Stepped Spillways

An experimental study on a large model flume with fiber-optical instrumentation indicated that minimum Reynolds and Weber numbers of about 105 and 100, respectively, are required for viscosity and surface tension effects to become negligible compared to gravitational and inertial forces expressed by Froude similitude. Both the location of and the flow depth at the inception point of air entrainment can be expressed as functions of a so-called roughness Froude number containing the unit discharge, step height and chute angle. The depth-averaged air concentration is found to depend only on a normalized vertical distance from the spillway crest and the chute angle for chute slopes ranging from embankment to gravity dam spillways. Air concentration profiles can be expressed by an air bubble diffusion model. The pseudobottom air concentration allows the assessment of the cavitation risk of stepped chutes and is approximated by a regression function. Finally, a new velocity distribution function is presented consisting of a power law up to 80% of the characteristic nondimensional mixture depth, and a constant value above.     

Characteristics of Skimming Flow over Stepped Spillways

This paper presents the results of a laboratory study on the characteristics of fully developed skimming flow in a large model of a stepped spillway for two slopes, for a range of discharges with yc∕h in the range of 0.7–4.4. Fully developed aerated flow on a stepped spillway can be divided into lower and upper regions, similar to those for self-aerated flow in steep chutes. The air concentration distributions in these two regions agree with the equations developed by Straub and Anderson for flow in steep chutes. It was found that the depth at which the air concentration is equal to 90% can be considered as the depth of aerated flow on stepped spillways. In the lower region, the velocity profiles were described by the Karman-Prandtl equation for rough turbulent flow when an equivalent bed roughness was used. A correlation was developed for the skin friction coefficient to predict the Reynolds shear stress at the virtual bed of the stepped spillway. It was found that the relative energy loss in the stepped spillway is in the range of 48–63%. It was also found that the mean air concentration on a stepped spillway is larger than that in a corresponding chute.     

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.     

Rough Wall Boundary Layer on Plates in Open Channels

Experiments were performed to measure the characteristics of a turbulent boundary layer developing on a rough surface placed in an open channel flow at close proximity to the free surface. Streamwise velocity measurements were made with a one-component laser Doppler velocimeter system at the top of the spherical roughness elements. Measurements at three stations downstream of the plate leading edge show the growth of the boundary layer on the rough wall and its interaction with the exterior open-channel flow and the free surface. Resorting to the turbulence profile provides an alternative definition of the boundary layer thickness. The near-wall flow follows the well-known logarithmic law with a shift due to roughness. In the outer layer, there are two opposing effects: the free surface tends to decrease the wake component while the roughness tends to increase it. The streamwise turbulence intensity is affected by the shear and turbulence in the exterior flow, the effect of the free surface being greater than that of wall roughness.     

Turbulent Boundary Layer Flows Above a Porous Surface Subject to Flow Injection

A new analytical expression for velocity profile in a fully developed turbulent boundary layer above a porous surface subject to flow injection is derived by solving the coupled Reynolds equations and turbulent kinetic energy equation. The advection of turbulent kinetic energy is considered during the derivation, whereas the earlier studies have neglected it. The new solution reduces to the universal logarithmic law in the case of no flow injection. For the small injection, the solution can be expanded into a series form in terms of the normalized injection velocity. The leading order terms are found to be equivalent to those in the earlier works in which the advection of turbulent kinetic energy has been neglected in the derivation. The new solution can provide more accurate prediction of bed shear stress for a wide range of flow injection rate, fluid type (e.g., from air to water), and surface roughness. On the other hand, the earlier theories may significantly underestimate bed shear stress under high injection rates.     

Developing Flow Region and Pressure Fluctuations on Steeply Sloping Stepped Spillways

The hydrodynamic pressure field is important for the design and safety of steeply sloping stepped spillways, which are typically designed for considerably lower maximum specific discharges than smooth spillways. The hydraulic performance of stepped spillways at high velocities may compromise its use due to major concern with safety against cavitation damage. Hydraulic model investigations were conducted in different large-size stepped chutes to characterize the nonaerated flow region which is potentially prone to cavitation damage and the pressure field acting on the step faces. The clear water depths and energy dissipation in the developing flow region are described in terms of integral measures of the turbulent boundary layer. Expressions for the location of and the flow depth at the inception point of air entrainment are derived. Pressure distributions on the horizontal and vertical faces of the step along the spillway are presented. Measurements indicated a different behavior of the pressure field in the aerated and nonaerated flow region. The mean and fluctuating pressure coefficients along the spillway are approximated by a regression function. The vertical face near the outer step edge close to the inception point of air entrainment is identified as a critical region for predicting cavitation inception in flow over stepped spillways. From the analysis of the pressure fluctuations in that region a maximum velocity of 15 m/s is proposed as a criterion to avoid extreme negative pressures in typical prototype steeply sloping stepped spillways, eventually leading to the occurrence of cavitation in the nonaerated flow.     

Turbulent Open-Channel Flow in Circular Corrugated Culverts

This paper presents the results of a laboratory study of the velocity field in turbulent open-channel flow in a circular corrugated pipe of diameter D of 0.622 m for three slopes S of 0.55, 1.14, and 2.55% and a range of discharges from 30 to 200 L∕s. The Manning n was found to be equal to 0.023. Velocities were relatively small in some portion of the flow near the boundary of the pipe, and these low velocity regions may be useful for fish passage upstream. In the region of fully developed flow, in the central vertical plane, the longitudinal velocity u was described by the Prandtl equation for rough turbulent flow, with a dip in the velocity profiles near the water surface. The velocity profiles in the noncentral planes were also described by the Prandtl equation for rough turbulent flow, but with a significant dip in the upper part of the flow. An empirical method was devised to describe the geometrical and kinematical properties of this velocity dip. The general findings of this study were also found to be valid for flow in a large corrugated pipe of diameter of 4.27 m with two slopes of 0.14 and 1.42%.     

Form Drag of Fences Placed in Disturbed Turbulent Boundary Layers

Wind tunnel studies on the drag characteristics of solid and porous fences placed in the turbulent boundary layer disturbed by the presence of an upstream fence are reported. The analysis of the data reveals that the outer region of the approach velocity profile also influences the drag, contrary to the implication from the relationship of Ranga Raju et al. that only the velocity distribution in the inner region is of consequence so far as the drag is concerned. A unique relationship has been obtained for the drag coefficient based on the average velocity over the height of the fence in terms of the porosity, the shape factor of the approach boundary layer, and an additional parameter characterizing the approach velocity distribution over the height of the fence. This relationship is valid for 2D solid and porous fences placed in a flow with a zero pressure gradient for the cases of disturbed and undisturbed boundary layers developed over smooth, transitional, and rough surfaces.     

Three-Dimensional Mean Velocity Analysis of a 30 Degree Bend Flow

The 3D velocity profiles of the 30° bend flow of Flack and Johnston have been analyzed in terms of the existing 3D turbulent boundary layer theories. Various cross-flow and near-wall similarity models were tested. Coles's cross-flow model described the velocity profiles satisfactorily. The wall function matched the data well, and the experimentally determined wake functions collapsed into a narrow band, which was however different from the originally suggested wake function. A new form of wake function has been proposed. Among the near-wall models, Hornung and Joubert's and Prahlad's models matched the data very well and excellent near-wall similarity from the wall to the boundary layer edge was achieved. This is rather unexpected in a 3D turbulent boundary layer flow with large skewing. The excellent performance of these two near-wall models could not be attributed to any particular reason.     

Effects of Bed Roughness on Flow around Bed-Mounted Cylinders in Open Channels

This paper presents the results of an experimental study of flow around cylindrical objects on a rough bed in an open channel. This is an extension of a previous study of flow around cylinders on a smooth bed. The purpose of this study is to explore the effects of bed roughness on the characteristics of the deflected flow around cylindrical objects and the resulting bed-shear stress distributions. Similar to the previous study cylindrical objects of equal diameter and four heights were tested under similar flow conditions producing four different levels of submergence. Bed shear stress and deflected flow velocities were measured by a thin yaw-type Preston probe after a set of flow visualization tests. Flow visualization tests showed that the horse-shoe vortex systems on the rough bed occupy a relatively greater width compared to the smooth bed. Unlike smooth bed observations, the flow separation point upstream of the cylinder was not dependent on the level of submergence as the separation points were found to appear within a short range of x = ?1D to ?1.2D. Bed shear stress has been found to increase significantly near the shoulder of the cylinders, and its ratio with respect to the approach bed-shear stress was twice as large compared to the smooth bed case. Mean velocity profiles were analyzed in terms of three-dimensional turbulent boundary layer theories. Bed roughness was found to oppose the effect of the lateral pressure gradient that causes skewing in the boundary layer. Perry and Joubert’s model has been found to be equally accurate on smooth and rough beds for predicting the deflected velocity magnitudes around cylinders. The present study will enhance the knowledge of hydraulics of flow around bed-mounted objects (e.g. fish-rocks) in natural streams.     

Plane Turbulent Wall Jets on Rough Boundaries with Limited Tailwater

Combining the results of a laboratory study of plane turbulent wall jets on rough boundaries with shallow tailwater, with the results of an earlier work of Rajaratnam on wall jets on rough boundaries with deep tailwater, this paper attempts to describe the effects of boundary roughness and tailwater depth on the characteristics of plane turbulent wall jets on rough beds, which are important in the field of hydraulic engineering. The time-averaged axial velocity profiles at different sections in the wall jet were found to be similar, with some difference from the profile of the classical plane wall jet. The normalized boundary layer thickness δ/b, where b is the length scale of the velocity profile, was equal to 0.35 for wall jets on rough boundaries compared to 0.16 for the classic wall jet. Two stages were seen to exist in the decay of the maximum velocity um as well as in the growth of the length scale, with the first stage corresponding to that of deep tailwater and the second stage to shallow tailwater. In the first stage, the decay of the maximum velocity um at any section in terms of the velocity u0 at the slot, with the longitudinal distance x in terms of L which is the distance where um = 0.5U0, was described by one general function, for smooth as well as rough boundaries. The length scale L in terms of slot width decreases as the relative roughness of the boundary increases. The onset of the second stage was not affected significantly by the bed roughness. The growth rate of the length scale b of the wall jet increased from 0.076 for a smooth boundary to about 0.125 for a relative roughness ks/b0 in the range of 0.25 to 0.50, where ks is the equivalent sand roughness and b0 is the thickness of the jet at the slot.     

Simulation of Free Surface Flow over Spillway

A numerical approach is proposed to simulate and study the effect of geometry on the free surface flow over a tunnel spillway. A three-step solution procedure is proposed to speed up the solution. The first step is to obtain an approximate free surface profile and mean velocity distribution, assuming 1D steady flow. Next, the 3D turbulent flow field is computed while the water surface profile is kept fixed. Finally, the water surface is set free to move and generate waves. The governing equations for weakly compressible flow (compressible hydrodynamic flow) are solved with an explicit finite volume method. A boundary fitted grid system is used to accurately resolve the flow near the free surface with steep waves. A mixed Lagrangian-Eulerian approach is proposed to calculate the new free surface position. The numerical results of a time-averaged free surface profile as well as pressure and velocity distribution have been compared with some experimental data.     

Experimental Study and 3D Numerical Simulations for a Free-Overflow Spillway

The main objectives of the present work were to investigate the flow field over a spillway and to simulate the flow by means of a three-dimensional (3D) numerical model. Depending on the wall curvature, the boundary layer parameters decreased or increased with increasing distance along the spillway. The growth of the boundary layer along the spillway is better described as a function of Reynolds number than the normalized streamwise length. A simplified form of the 3D momentum equation can be used to obtain a rough estimate of the skin friction. The velocity profile in the boundary layer along the spillway is described by a velocity–defect relationship. Numerical models provide a cost-effective means of simulating spillway flows. In this study, the water surface profiles and the discharge coefficients for a laboratory spillway were predicted within an accuracy range of 1.5–2.9%. The simulations were sensitive to the choice of the wall function, grid spacing, and Reynolds number. A nonequilibrium wall function with a grid spacing equal to a distance of 30 wall units gave good results.     

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.     

Skimming Flow in the Nonaerated Region of Stepped Spillways over Embankment Dams

Traditionally, research on stepped spillway hydraulics has been focused on the air-water flow region but for the hydraulic design of small embankment dams experiencing relatively large overtopping flows, the nonaerated region can be very important. Empirical formulas are presented for predicting skimming flow properties upstream of the point of inception of air entrainment for 1V:2H sloping stepped spillways, and the location and flow depth at the point of inception. Particular emphasis is placed on the clear-water depth, velocity distribution, and the energy dissipation characteristics in the developing nonaerated flow region. The velocity distribution is well described by a power law. The normalized clear-water depth and the normalized specific energy varied with the relative distance along the spillway and the effect of the normalized critical depth was negligible. Finally, the rate of energy dissipation was small, which has direct implications for the design of the downstream energy dissipator.     

Large-Eddy Simulation of Turbulent Flow over a Fixed Two-Dimensional Dune

Turbulent open-channel flow over a two-dimensional dune is studied using an established large-eddy simulation code. The free surface is approximated as a shear free boundary. Turbulence statistics and instantaneous flow structures are examined. Numerical results from two computational grids agree with each other, and are also in good agreement with recently obtained experimental data. The mean velocity profiles show significant changes along the dune and there is no region that conforms to the standard law-of-the-wall. Profiles of the Reynolds stresses show distinct peaks marking the shear layer that originates from flow separation at the dune crest. Secondary peaks found further from the dune are ascribed to the shear layer over the upstream dune. Details of the separated flow and development of the flow after reattachment are well predicted. Quadrant analysis of the Reynolds shear stress shows that turbulent ejections dominate the near-wall motions. Complex water surface flow structures are visualized.     

Turbulent Boundary Layer over Symmetric Bodies with Rigid and Flexible Surfaces

This paper presents the results of an investigation concerning the development of a turbulent boundary layer over a 2D symmetrical aerofoil and a 3D axisymmetric body with rigid and flexible surfaces. The experimental work included detailed measurements of the mean velocity profiles, pressure distribution, and drag force. The thin shear layer equations were solved numerically using a modified turbulence model to obtain the characteristics of the turbulent boundary layer. The results of this study indicate a significant difference between the characteristics of flow over rigid surfaces and those of flow over flexible surfaces of the same geometry. The mean velocity of flow in the case of flexible surfaces is smaller than the corresponding velocity of flow in the case of a rigid surface for a major part of the boundary layer. The boundary layer thicknesses are consistently higher on flexible surfaces than those on the corresponding rigid surfaces. Furthermore, in the case of flexible surfaces, drag reduction was always observed. The amount of reduction was seen to be systematically dependent on the characteristics of the flexible surface.     

Outer Scaling for Open Channel Flow over a Gravel Bed

Similarity analysis is performed for hydraulically rough open channel flow over a gravel bed to provide mixed outer scaling of the mean-velocity profile. The analysis is based on equilibrium turbulent boundary-layer theory derived using the asymptotic invariance principle. Outer scaling based on the similarity theory is validated with velocity measurements from the laboratory and field, having a Reynolds number range that includes 1×104, 1×105, and 1×106 and a Froude number range from 0.26 to 0.83. The results show that the free-stream velocity is an appropriate outer scale for gravel-bed river flows at moderate and bankfull stage. The results agree well with the velocity measurements and collapse laboratory and field data, which allow an important connection between open channel research in the laboratory and the applications for which the research is performed in the field. The results show that the R/aD84 roughness parameter is consistent with the mixed scale used in boundary-layer velocity scaling. This is in agreement with the consistent turbulent structure of the flow for both flat plate boundary-layer and open channel flow scenarios. While R/D84 has been used empirically with depth-averaged velocity and roughness laws for many years, this roughness parameter is shown in a theoretical context due to its influence on the turbulent structure of the flow. The results are applicable to modeling the velocity distribution under fundamental gravel-bed flow cases that span to the bankfull flow regime, which provides a contribution to stream engineering.     

Flow Measurement Using Flying ADV Probes

An innovative measurement system of “flying” acoustic Doppler velocimeters was designed in order to allow rapid velocity measurement over a large flow field. Such measurements are necessary, for example, when measuring over a temporally varying and locally nonuniform rough bed. The measurement technique was verified by comparison with measurements taken in the same flows using a traditional stationary probe technique. Comparison showed that the flying-probe approach performs similarly to stationary measurements in capturing the mean flow field and turbulent fluctuations. The data obtained from flying probe experiments can be used to describe the flow in terms of double-averaged hydrodynamic variables, obtained by averaging in time and spatial domains within a thin slab parallel to the mean bed. Examples are presented of flow measurements over a fixed flat bed, a fixed dune bed, and over mobile developing bed forms. It is shown that near-bed measurements suffer from boundary reflection interference, though affected data can be filtered out based on the ADV-measured correlation coefficient. Measurement below roughness tops is possible, with in-bed records being detectable by spikes in measured signal-to-noise-ratio and by comparison with measured bed topography.