Open Boundary and Frictional Influences in 3D Tidal Models

Both 2D and 3D hydrodynamic models of the Irish Sea are used to examine the influence of the open boundary condition (OBC), bottom friction, and vertical eddy viscosity upon tidal currents in this region. Three eddy viscosity formulations, namely, a two-equation turbulence model, a simple flow-dependent eddy viscosity model, and a constant eddy viscosity of 1,000 cm2∕s, are considered. In an initial series of 2D calculations the influence of OBC and bottom friction coefficient (BFC) upon tidal elevations is examined with a view to determining an optimal value related to different OBCs, in the sense that the model can reproduce the extensive data set of elevations in the region. In subsequent 3D calculations in which bottom friction is related to the depth-mean current, the influence of vertical eddy viscosity upon computed currents is examined using identical BFCs to those used in the 2D model. These calculations clearly show the performance of different vertical eddy viscosity formulations in a 3D model. Calculations are also carried out using a single point model in the vertical. Results from the calculations and comparisons with data show that the optimal BFC can vary depending upon the vertical eddy viscosity formulation used in the calculation. Computed elevations and currents from a 3D model (in which bottom friction is computed from bottom current) are compared with observations and show a similar agreement for each viscosity parameterization provided the optimal BFC for this parameterization is used. These calculations suggest that to rigorously test a range of turbulence models in 3D calculations, in addition to accurate current measurements in the bottom boundary layer, it is also necessary to determine the bed types and forms and also to derive accurate tidal input to the model along its open boundaries.     

Bed Drag Coefficient Variability under Wind Waves in a Tidal Estuary

In this paper we report the results of a study of the variation of shear stress and the bottom drag coefficient CD with sea state and currents at a shallow site in San Francisco Bay. We compare shear stresses calculated from turbulent velocity measurements with the model of Styles and Glenn reported in 2000. Although this model was formulated to predict shear stress under ocean swell on the continental shelf, results from our experiments show that it accurately predicts these bottom stress under wind waves in an estuary. Higher up in the water column, the steady wind-driven boundary layer at the free surface overlaps with the steady bottom boundary layer. By calculating the wind stress at the surface and assuming a linear variation of shear between the bed and surface, however, the model can be extended to predict water column shear stresses that agree well with data. Despite the fidelity of the model, an examination of the observed stresses deduced using different wave–turbulence decomposition schemes suggests that wave–turbulence interactions are important, enhancing turbulent shear stresses at wave frequencies.     

Depth-Averaged Drag Coefficient for Modeling Flow through Suspended Canopies

Aquatic suspended canopies are porous obstacles that extend down from the free-surface but have a gap between the canopy and bed. Examples of suspended canopies include those formed by aquaculture structures or floating vegetation. The major difference between suspended canopies and the more common submerged canopies, which are located on the bottom boundary, is the influence of the bottom boundary layer beneath the suspended canopy. Data from laboratory experiments are presented which explore aspects of the flow through and beneath suspended canopies constructed from rigid cylinders. The experiments, using both acoustic Doppler and two-dimensional (2D) particle tracking velocimetry, give details of the flow structure that may be divided vertically into a bottom boundary layer, a canopy shear layer, and an internal canopy layer. The experimental data show that the penetration of the shear layer into the canopy is limited by the distance between the canopy and bottom boundary layer. Peaks in velocity spectra indicate an interaction between the bottom boundary and canopy shear layer. An analytical model is also developed that can be used to calculate a drag coefficient that includes the effect of both canopy drag and bed friction. This drag coefficient is suitable for use in 2D (depth-averaged) hydrodynamic modeling. The model also allows the average velocity within and beneath the canopy to be calculated, and is used to investigate the effect of canopy density and thickness on both total drag and bottom friction.     

Momentum Exchange in Straight Uniform Compound Channel Flow

Transverse exchange of momentum between the channel and the floodplain in straight uniform compound channel flow is considered in this paper. This process results in the so-called “kinematic effect,” a lowering of the total discharge capacity of a compound channel compared to the case where the channel and the floodplain are considered separately. The mechanisms responsible for the momentum exchange are considered. The transverse shear stress in the mixing region is modeled using a newly developed effective eddy viscosity concept, that contains: (1) the effects of horizontal coherent structures moving on an uneven bottom, taking compression and stretching of the vortices into account and (2) the effects of the three-dimensional bottom turbulence. The model gives a good prediction of the transverse profiles of the streamwise velocity and the transverse shear stress of the flood channel facility experiments. Characteristic features of the lateral profile of the eddy viscosity are also well predicted qualitatively, but in a quantitative sense there is room for improvement. Secondary circulations are shown to be of minor importance in straight uniform compound channel flows.     

Laminar Water Wave and Current Passing Over Porous Bed

The problem of the dynamic interaction of water waves, current, and a hard poroelastic bed is dealt with in this study. Finite-depth homogeneous water with harmonic linear water waves passing over a semi-infinite poroelastic bed is investigated. In order to reveal the importance of viscous effect for different bed forms, viscosity of water is considered herein. In a boundary layer correction approach, the governing equations of the poroelastic material are decoupled without losing physical generality. The contribution of pressure effect and shear effect to the hard poroelastic bed, which is a valuable indication to the mechanism of ripple formation, is clarified in the present study. This approach will be helpful in saving time and storage capacity when it is applied to numerical computation.     

Bed Shear Stress Boundary Condition for Storage Tank Sedimentation

Computational fluid dynamics-based (CFD) software tools enable engineers to simulate flow patterns and sediment transport in ancillary structures of sewer systems. Lagrangian particle tracking represents a computationally efficient technique for modeling sediment transport. In order to represent the process of sedimentation in storage tanks, careful consideration must be given to the boundary condition at the bottom of the tanks. None of the boundary conditions currently available in the FLUENT CFD software appears to represent the observed behavior of sediment particles, which may become resuspended after first contact with the bed if the local flow velocity is sufficiently high. In this study, a boundary condition based on bed shear stress has been implemented in FLUENT and evaluated against laboratory data. A particle is trapped if the local bed shear stress is below the critical bed shear stress; otherwise, the particle is resuspended. The approach gives satisfactory agreement with measured sedimentation efficiency data, and the simulated spatial distribution is very similar to the sediment distribution observed in a laboratory tank.     

Predicting Boundary Shear Stress and Sediment Transport over Bed Forms

To estimate bed-load sediment transport rates in flows over bed forms such as ripples and dunes, spatially averaged velocity profiles are frequently used to predict mean boundary shear stress. However, such averaging obscures the complex, nonlinear interaction of wake decay, boundary-layer development, and topographically induced acceleration downstream of flow separation and often leads to inaccurate estimates of boundary stress, particularly skin friction, which is critically important in predicting bed-load transport rates. This paper presents an alternative methodology for predicting skin friction over 2D bed forms. The approach is based on combining the equations describing the mechanics of the internal boundary layer with semiempirical structure functions to predict the velocity at the crest of a bedform, where the flow is most similar to a uniform boundary layer. Significantly, the methodology is directed toward making specific predictions only at the bed-form crest, and as a result it avoids the difficulty and questionable validity of spatial averaging. The model provides an accurate estimate of the skin friction at the crest where transport rates are highest. Simple geometric constraints can be used to derive the mean transport rates as long as bed load is dominant.     

Shear Stress in Smooth Rectangular Open-Channel Flows

The average bed and sidewall shear stresses in smooth rectangular open-channel flows are determined after solving the continuity and momentum equations. The analysis shows that the shear stresses are function of three components: (1) gravitational; (2) secondary flows; and (3) interfacial shear stress. An analytical solution in terms of series expansion is obtained for the case of constant eddy viscosity without secondary currents. In comparison with laboratory measurements, it slightly overestimates the average bed shear stress measurements but underestimates the average sidewall shear stress by 17% when the width–depth ratio becomes large. A second approximation is formulated after introducing two empirical correction factors. The second approximation agrees very well (R2>0.99 and average relative error less than 6%) with experimental measurements over a wide range of width–depth ratios.     

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.     

Hydraulics of Submerged Jet Subject to Change in Cohesive Bed Geometry

The results of an experimental investigation of the time variation of scour hole and the flow characteristics of the quasi-equilibrium state of scour of a cohesive bed downstream of an apron due to a submerged horizontal jet issuing from a sluice opening are presented. Experiments were carried out with natural cohesive sediment for various sluice openings, jet velocities, and lengths of apron. Attempts are made to explain the similarity existing either in the process of scour or in the scour profiles that the scour holes follow downstream of an apron. The scour profiles at different times follow a particular geometrical similarity and can be expressed by a polynomial using relevant parameters. The characteristic parameters affecting the time variation of scour depth are identified based on the physical reasoning and dimensional analysis. An equation for time variation of maximum scour depth is obtained empirically. The diffusion characteristics of the submerged jet, growth of boundary layer thickness, velocity distribution within the boundary layer, and shear stress at the quasi-equilibrium state of scour are also investigated. The expression of shear stress is obtained from the solution of the von Kármán momentum integral equation.     

Boundary Shear Stress in Spatially Varied Flow with Increasing Discharge

The distribution of the wall shear stress on the bed and sidewalls of an open channel receiving lateral inflow was obtained from experimental measurements of the distribution of the velocity in the viscous sublayer using a laser doppler velocimeter. The experiments were conducted in a 0.4 m wide by 7.5 m long flume. Lateral inflow was provided into the channel from above via sets of nozzles positioned toward the downstream end of the flume. Lateral inflow was provided over a length of 1.9 m. The results indicate that the local boundary shear stresses are significantly influenced by lateral inflow. The significant variation occurs near and around the region where the lateral inflow enters the channel. At various measurement positions along the lateral inflow zone, mean boundary, mean wall, and mean bed shear stresses were obtained and compared. The results indicate that the mean boundary shear stresses increase from the upstream to the downstream ends of the lateral inflow zone. The results also indicate that the mean bed shear stress is always greater than the mean wall shear stress, which are approximately 30–60% of the mean bed shear stress. The friction factor in the Darcy–Weisbach equation was obtained from both the mean boundary shear stress and from the equation describing the water surface elevation in an open channel receiving lateral inflow (equation for spatially varied flow with increasing discharge). The results indicate that the estimated friction factors from the latter approach are significantly larger. Also, the estimated friction factors from both approaches are higher than the values predicted from the Blasius equation which describes the friction factor for wide uniform open channel flows. They were also higher than values predicted from the Keulegan equation, which is an empirically derived equation for flow in roof drainage gutters. The study highlights the deficiencies in the existing equations used to predict friction factors for spatially varied flow and that further research is required to explore the distribution of boundary shear stress in an open channel receiving lateral inflow.     

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.     

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.     

Periodic Diffusional Mass Transfer near Sediment/Water Interface: Theory

Time-variable (periodic) flow over a lake bed, and the associated boundary layer development, have the potential to control or at least influence rates of mass transfer across the sediment/water interface. An analysis for instantaneous and time averaged flux of a material across the sediment/water interface for infinite supply in the water and infinite sink in the sediment is presented. The water flow above the interface is characterized by the shear velocity (U?) which is a periodic function of time with a maximum amplitude of (U?0) as may be typical of an internal seiche (internal standing wave) motion in a density stratified lake. The relationship between the shear velocity on the lake bed and the wind shear on the lake surface is illustrated for an extremely simplified two-layered lake of constant depth. For a less restrictive analysis, shear velocities on a lake bed have to be obtained either from field measurements or from a three-dimensional lake circulation model driven by atmospheric forcing including wind. Smaller and wind-sheltered lakes will have lower (U?0) and periodicities (T). The response of the diffusive boundary layer was related to the period of the periodic motion (T), Schmidt number (Sc), and shear velocity (U?). The vertical diffusive flux at the sediment/water interface was expressed by a Sherwood number (Sh), either instantaneous or time averaged. The mean Sherwood number (Shave) varies with shear velocity of the wave motion over the sediment bed, Schmidt number (Sc) and the period (T) due to the response of the diffusive boundary layer to the time variable water velocity. Effective diffusive boundary layers develop only at low shear velocities. Where they do, maximum and minimum boundary layer thickness depends on all three independent variables (T, Sc, and U?0). The diffusive boundary layer strongly affects sediment/water mass transfer, i.e., Sherwood numbers. Mass transfer averaged over a period can be substantially less than that produced by steady-state flow at the same U?0 and Sc. At Sc = 500, typical for dissolved oxygen, the mass transfer ratio can be reduced to 60% of steady state, depending on the internal wave period (T).     

Structure of Turbulent Flow in a Shallow Tidal Estuary

A high-resolution current profiler (HRCP), which belongs to a pulse-to-pulse coherent Doppler sonar, has been used to measure vertical profiles of turbulence parameters, such as the Reynolds stresses, eddy viscosity, production and dissipation rates, etc., and to test the parametrization of dissipation rate and eddy viscosity. The HRCP and automatic ascending/descending CTD are deployed during the autumn of 2001 for 24 h in a tidal estuary. Reliable velocities along the beams with HRCP are collected with 3 s intervals and a vertical resolution as fine as 0.03 m in the range 0.02–0.98 m above the bottom. Density profiles with the CTD are taken nominally every 30 sec. The turbulent velocity variables depend largely on the tidal phase; the variables during the ebb deviate from those in neutral equilibrium boundary layer. This deviation during the ebb presumably arises from the “inactive motion.” The stability function SM in the Mellor–Yamada (M–Y) model is smaller than 0.39 even when the stratification is negligible during the flood. The constant of proportionality B1 in the dissipation model is larger than 16.6 used in M–Y model. There is room for improving some of the mixing parametrizations in estuarine tidal flows.     

Oxygen Demand by a Sediment Bed of Finite Length

A model of sedimentary oxygen demand (SOD) for a sediment bed of finite length is presented. The responses of diffusive oxygen transfer in turbulent flow above the sediment surface and of microbial activity inside the sediment to a developing diffusive boundary layer are modeled numerically. The developing diffusive boundary layer above the sediment/water interface is modeled based on shear velocity and turbulent boundary layer concepts, and dissolved oxygen (DO) uptake inside the sediment is modeled as a function of the microbial growth rate. The model predicts that the diffusive boundary layer above the sediment/water interface thickens in flow direction, and that DO penetration depth into the sediment is practically constant over the length of the sediment bed. The effect of the developing diffusive boundary layer on SOD is minor, except at very low shear/flow velocities (shear velocity U*<0.01?cm/s) and/or high microbial density inside the sediment. The average SOD over the sediment bed therefore varies only slightly with its length. SOD varies somewhat in flow direction, i.e., SOD is largest near the leading edge (x = 0), decreases with distance, and finally, approaches a nearly constant value for fully developed boundary layer. Including microbial activity in the sediment makes the change of SOD in flow direction much smaller than is predicted by a pure vertical diffusive flux model. The diffusive boundary layer is nearly fully developed at a dimensionless distance x+ = 10,000, regardless of microbial activity inside the sediment. Longer sediment beds are required to eliminate the small leading edge effect on any measured average SOD value. SOD depends strongly on the diffusion coefficient of DO inside the sediment bed. This effect becomes more significant as shear/flow velocity is increased. Overall, SOD is found to be controlled principally by shear velocity of the water flowing above the sediment/water interface, microbial activity inside the sediment, and diffusion of DO inside the sediment. The length of the sediment bed is of lesser influence.     

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.     

Momentum Transport in Sharp Open-Channel Bends

Flow in open-channel bends is characterized by cross-stream circulation, which redistributes the velocity and the boundary shear stress and thereby shapes the characteristic bed topography. Besides a center-region cell, classical helical motion, a weaker counterrotating outer-bank cell often exists. In spite of its engineering importance, the mechanisms underlying distributions of the velocity and the boundary shear stress in open-channel bends, and especially the role of both circulation cells, are not yet fully understood. In order to investigate these mechanisms, an evaluation is made of the various terms in the momentum equations based on the data measured, which gave the following results. The outer-bank cell forms a buffer layer that protects the outer bank from any influence of the center-region cell and keeps the core of maximum velocity a distance from the bank. Advective momentum transport by the center-region cell is a dominant mechanism; it significantly contributes to the observed outward shift of the downstream velocity and the bed shear stress and to flattening of the vertical profiles of the velocity. This important advective momentum redistribution has to be included in the depth-integrated flow models often used in engineering practice. Commonly used linear models overpredict the effects of the center-region cell. Based on results of the analysis of experimental data, these models are extended by accounting for the feedback between the center-region cell and the downstream velocity. The nonlinear model obtained clearly reveals the mechanisms of the center-region cell and its advective momentum transport. An analysis of nonlinear model results confirms and complements the analysis of experimental data. A true quasithree-dimensional flow model is obtained by coupling this nonlinear model to depth-integrated flow models, thus providing an engineering tool for morphodynamical investigations.     

Turbulent Stresses at the Bottom Surface near an Abutment: Laboratory-Scale Numerical Experiment

The flow field around a bridge abutment is analyzed by means of large eddy simulation. The geometrical configuration corresponds to the initial condition of a scour process (flat bed). The three-dimensional flow structure in front of the abutment is analyzed with special emphasis on its effects on shear stresses and pressure gradients on the bottom wall which, in turn, are discussed with respect to their potential scouring action. Both first- and second-order statistics around the abutment are quantitatively discussed, together with probability density distributions of stresses in specific locations. The investigation shows that several terms may play a relevant role in sediment transport around the obstacle. Specifically, the mean horizontal pressure gradient may reach values as large as two orders of magnitude that of a canonical boundary layer, whereas the instantaneous vertical pressure gradient may give an uplifting force comparable to the immersed weight of the sediment. The analysis suggests that local scour models should incorporate the contribution to the destabilizing force coming from pressure stresses and from turbulent fluctuations.     

Simplified Model of Tractive-Force Distribution in Closed Conduits

A simplified model for the computation of boundary shear stress distributions acting on the flow perimeter of closed ducts is presented. The model assumes that the surplus energy within any control volume in a three-dimensional flow will be transferred towards the nearest boundary to be dissipated. Based on this model, the flow cross sectional area in the closed duct is divided into subflow regions corresponding to the side walls and bed, and the shear distributions over the wetted perimeter within each subflow area are assessed. Analytical equations, valid for all channel aspect ratios, for the prediction of local and mean shear stresses along the bed and side walls in smooth rectangular duct flow are derived. The formulae give good predictions of the shear stress distributions when compared with existing experimental data in the literature. The possible applications of the model to nonrectangular duct flows are also discussed.