Bed Stress Research Articles

Large eddy simulation of a long asymmetric compound open channel

A detailed Large Eddy Simulation of turbulent flow in a long compound open channel with one floodplain is reported. The Reynolds number is approximately 42,000 and the free surface was treated as fully deformable. The results are in agreement with experimental measurements and support the use of high spatial resolution and a large box length in contrast with a previous simulation of the same geometry. Details are reported of the time-varying floodplain flow and flow at the junction of the channel and the floodplain.A secondary flow is identified at the internal corner that persists and increases the bed stress on the floodplain. This phenomenon results from the lateral momentum transfer from the main channel by the extended secondary circulation existing in the interface region. This transfer of fluid is dominated by the relatively intense vorticity generated at the interface wall and corner. The regions of highest turbulent activity are elongated streaks near the walls that are aligned with the mean flow.

Sandbar formation under surface waves: Theory and experiments

We report a combined theoretical and experimental study of sandbar formation under simple‐harmonic surface waves. For coarse grains and weak waves, an established empirical rule of bed load transport is used with an asymptotic theory for the fluid flow. The surface waves are governed by potential theory and a depth‐linear eddy viscosity is employed in the turbulent boundary layer at the seabed. The derived bed stress is used to predict the sand‐bed evolution. Laboratory experiments and corresponding numerical simulations for both high and low beach reflection are discussed. For weak reflection, the shear stress associated with the return current is found to be important. Partial simulation of a field record in Cape Cod Bay is also described.

A Note on the Effect of Wind Waves on Vertical Mixing in Franks Tract, Sacramento–San Joaquin Delta, California

A one-dimensional numerical model that simulates the effects of whitecapping waves was used to investigate the importance of whitecapping waves to vertical mixing at a 3-meter-deep site in Franks Tract in the Sacramento-San Joaquin Delta over an 11-day period. Locally-generated waves of mean period approximately 2 s were generated under strong wind conditions; significant wave heights ranged from 0 to 0.3 m. A surface turbulent kinetic energy flux was used to model whitecapping waves during periods when wind speeds > 5 m s-1 (62% of observations). The surface was modeled as a wind stress log-layer for the remaining 38% of the observations. The model results demonstrated that under moderate wind conditions (5–8 m s-1 at 10 m above water level), and hence moderate wave heights, whitecapping waves provided the dominant source of turbulent kinetic energy to only the top 10% of the water column. Under stronger wind (> 8 m s-1), and hence larger wave conditions, whitecapping waves provided the dominant source of turbulent kinetic energy over a larger portion of the water column; however, this region extended to the bottom half of the water column for only 7% of the observation period. The model results indicated that phytoplankton concentrations close to the bed were unlikely to be affected by the whitecapping of waves, and that the formation of concentration boundary layers due to benthic grazing was unlikely to be disrupted by whitecapping waves. Furthermore, vertical mixing of suspended sediment was unlikely to be affected by whitecapping waves under the conditions experienced during the 11-day experiment. Instead, the bed stress provided by tidal currents was the dominant source of turbulent kinetic energy over the bottom half of the water column for the majority of the 11-day period.

Erosion of a granular bed driven by laminar fluid flow

Motivated by examples of erosive incision of channels in sand, we investigate the motion of individual grains in a granular bed driven by a laminar fluid to give us new insights into the relationship between hydrodynamic stress and surface granular flow. A closed cell of rectangular cross-section is partially filled with glass beads and a constant fluid fluxQflows through the cell. The refractive indices of the fluid and the glass beads are matched and the cell is illuminated with a laser sheet, allowing us to image individual beads. The bed erodes to a rest heighthrwhich depends onQ. The Shields threshold criterion assumes that the non-dimensional ratio θ of the viscous stress on the bed to the hydrostatic pressure difference across a grain is sufficient to predict the granular flux. Furthermore, the Shields criterion states that the granular flux is non-zero only for θ > θc. We find that the Shields criterion describes the observed relationshiphr∝Q1/2when the bed height is offset by approximately half a grain diameter. Introducing this offset in the estimation of θ yields a collapse of the measured Einstein numberq* to a power-law function of θ − θcwith exponent 1.75 ± 0.25. The dynamics of the bed height relaxation are described well by the power-law relationship between the granular flux and the bed stress.

Observations of sediment transport on the continental shelf off the mouth of the Waiapu River, New Zealand: Evidence for current-supported gravity flows

Instrumented tripods deployed at depths of 40 and 60 m on the shelf off the mouth of the Waiapu River on the east coast of New Zealand's North Island recorded data on waves, currents, and sediment fluxes from May 22 to August 10, 2004. Three major flood events and several wave events occurred during the deployment. Data from acoustic Doppler velocimeters and profilers revealed that downslope sediment fluxes accompanied a flood event of late June, during which near-bed downslope current speeds approached 0.5 m s −1. The most pronounced downslope transport within the benthic layer occurred the day after peak flood but coincided with strong isobath-parallel currents. Suspended sediment concentrations about 1 m above the bed were on the order of 2–4 g l −1 at times of maximum seaward flow. Suspension of freshly discharged sediment within the hyperpycnal layer over much of the profile was maintained by the high bed stresses associated with the strong benthic currents, in contrast to the wave-supported sediment gravity flows recently reported on other river-nourished shelf systems. Nonetheless, observed concentrations and velocities were largely consistent with the dynamics of critically stratified sediment gravity flows based on equations previously applied elsewhere to wave-supported cases. Despite lower concentrations in current-supported gravity currents, the greater thickness results in similar total loads. In further contrast to earlier results, our data suggest that as sediment off the Waiapu flowed into deeper water across the seaward steepening bottom profile, auto-suspension may have aided in thickening and accelerating the gravity current.

Time-averaged near-bed suspended sediment concentrations under waves and currents: Comparison of measured and model estimates

Profiles of suspended sediment concentration and velocity were measured over a 15-day period at a near-shore site off Santa Cruz, CA in Monterey Bay. The concentration and velocity data were collected with an Acoustic Backscattering System (ABS) and Acoustic Current Profiler (ACP) that were mounted on a bottom tripod. High-resolution bottom scanning sonar was also attached to the tripod to provide images of bed features during the experiment. Hourly time-averaged near-bed concentrations of suspended sediment were calculated from three models and compared with the measurements. Surface waves and currents that were generated by a storm of moderate intensity caused bed stresses that exceeded threshold stress for D 50=0.02 cm, the median size of the moderately well-sorted bottom sediment, over a period of about 7 days. Estimates of the concentration at 1 cm above the bottom, C a1, were obtained using the ABS measurements. These observations have been compared with predictions for the concentration at 1 cm above the bottom, C 1. Nielsen's models for reference concentration C o [Nielsen, P., 1986. Suspended sediment concentrations under waves. Coastal Engineering 10, 32–31; Nielsen, P., 1992. Coastal Bottom Boundary Layers and Sediment Transport, Advanced Series on Ocean Engineering. World Scientific, Hackensack, NJ.] are purely wave-based and do not include effects of bottom currents on bed stress and bedform scales. C 1 calculated from this model compared well with measured C a1 when currents were weak and small oscillatory ripples were observed in the sonar images. However, during the 3-day period of highest bottom stresses modeled C 1 did not compare well to C a1. The other two models for C 1, Glenn and Grant [Glenn, S.M., Grant, W.D., 1987. A suspended sediment stratification correction for combined wave and current flows. Journal of Geophysical Research 92(C8), 8244–8264.] and van Rijn and Walstra [Van Rijn, L.C., Walstra, D.J.R., 2004. Description of TRANSPOR2004 and implementation in Delft3D-ONLINE. Interim Report prepared for DG Rijkswaterstaat, Rijksinstituut voor Kust en Zee. Delft Hydraulics Institute, The Netherlands.], accounted for combined wave-current stresses and included different formulations for predicting bedform scales. C 1 calculated from these models compared well with measurements throughout the early and middle period of the storm, but underpredicted measured values during the latter part of the experiment. An alternative method based on dimensional analysis provided a straightforward way to relate C 1 to bed stress and sediment parameters. This new relationship based on dimensional analysis explained about 77% of the variability in C a1 derived from the ABS data.

Sediment pumping by tidal asymmetry in a partially mixed estuary

Observations collected at two laterally adjacent locations are used to examine the processes driving sediment transport in the partially mixed York River Estuary. Estimates of sediment flux are decomposed into advective and pumping components, to evaluate the importance of tidal asymmetries in turbulent mixing. At the instrumented location in the estuarine channel, a strong asymmetry in internal mixing due to tidal straining is documented, with higher values of eddy viscosity occurring during the less‐stratified flood tide. As a result of this asymmetry, more sediment is resuspended during the flood phase of the tide resulting in up‐estuary pumping of sediment despite a net down‐estuary advective flux. At the instrumented location on the adjacent shoal, where no pronounced tidal asymmetry in internal mixing was found, both the pumping flux and advective flux were directed down‐estuary. The down‐estuary pumping of sediment on the shoal appears to be driven by asymmetries in bed stress. The impact of tidal asymmetries in bed stress at the channel location was negated because the amount of sediment available for resuspension was limited. As a result, the pumping flux was dominated by the overlying asymmetries in internal mixing. The asymmetries in stratification appear to exert an important control on the vertical distribution of sediment by both impacting the eddy diffusivity as well as the fall velocity. During the more turbulent flood tide, the fall velocities are smaller suggesting the Kolmogorov microscale is setting the upper bound on floc diameter.

Jet and hydraulic jump near‐bed stresses below a horseshoe waterfall

Horseshoe waterfalls are a common but unexplained feature of bedrock rivers. As a step toward understanding their geomorphology, a six‐component force/torque sensor was used to evaluate clear‐water scour mechanisms in a 0.91‐m broad‐crested horseshoe waterfall built into a 2.75‐m‐wide flume to measure longitudinal, lateral, and temporal profiles of the near‐bed spatially averaged stress vector. Jet impact caused moderate near‐bed stress. Downstream of the impact point, the stress vector shifted into a much more forceful upthrust with peak stress beyond the boil crest. Instantaneous fluctuations in near‐bed lift exceeded drag by 5–10 times and occurred over a wider range of frequencies, suggesting a larger role for particle entrainment. Predictions of bed stress significantly underestimated actual conditions. These results show that horseshoe waterfalls have an extended region of high scour along their centerline corresponding with the three‐dimensional convergent dynamics associated with both jet impact and jump‐enhanced lift fluctuations.

Energy Transport by Nonlinear Internal Waves

Abstract Winter stratification on Oregon’s continental shelf often produces a near-bottom layer of dense fluid that acts as an internal waveguide upon which nonlinear internal waves propagate. Shipboard profiling and bottom lander observations capture disturbances that exhibit properties of internal solitary waves, bores, and gravity currents. Wavelike pulses are highly turbulent (instantaneous bed stresses are 1 N m−2), resuspending bottom sediments into the water column and raising them 30+ m above the seafloor. The wave cross-shelf transport of fluid often counters the time-averaged Ekman transport in the bottom boundary layer. In the nonlinear internal waves that were observed, the kinetic energy is roughly equal to the available potential energy and is O(0.1) megajoules per meter of coastline. The energy transported by these waves includes a nonlinear advection term 〈uE〉 that is negligible in linear internal waves. Unlike linear internal waves, the pressure–velocity energy flux 〈up〉 includes important contributions from nonhydrostatic effects and surface displacement. It is found that, statistically, 〈uE〉 ≃ 2〈up〉. Vertical profiles through these waves of elevation indicate that up(z) is more important in transporting energy near the seafloor while uE(z) dominates farther from the bottom. With the wave speed c estimated from weakly nonlinear wave theory, it is verified experimentally that the total energy transported by the waves is 〈up〉 + 〈uE〉 ≃ c〈E〉. The high but intermittent energy flux by the waves is, in an averaged sense, O(100) watts per meter of coastline. This is similar to independent estimates of the shoreward energy flux in the semidiurnal internal tide at the shelf break.

Observations of wave orbital scale ripples and a nonequilibrium time‐dependent model

Measurements of seafloor ripples under wave‐dominated conditions from the LEO15 site and the Martha’s Vineyard coastal observatory were used to develop a time‐dependent model for ripple geometry. The measurements consisted of backscatter imagery from rotary side‐scan sonars, centimeter resolution bathymetric maps from a two‐axis rotary pencil‐beam sonar, and forcing hydrodynamics. During moderate energy conditions the ripple wavelength typically scaled with wave orbital diameter. In more energetic conditions the ripples reached a maximum wavelength of 0.8 to 1.2 m and did not continue to increase in wavelength or decrease in height. The observations showed that the relict ripples left after storms typically had wavelengths close to the maximum wavelength. The time‐dependent model is based on an equilibrium model that allows the ripples to maintain wavelength proportional to wave orbital diameter until a suspension threshold determined by wave velocity and grain size is reached. The time‐dependent model allows the ripple spectra to follow the equilibrium solution with a temporal delay that is based on the ratio of the ripple cross‐sectional area to the sediment transport rate. The data was compared to the equilibrium model, a simplified version of the time‐dependent model (where the ripples were assumed to follow the equilibrium model only when the bed stress was sufficient to move sediment), and the complete time‐dependent model. It was found that only the complete time‐dependent model was able to correctly predict the long wavelength relict ripples and that the other approaches underpredicted relict ripple wavelengths.

Numerical Simulations of the Flow and Sediment Transport Regimes Surrounding a Short Cylinder

The 3-D flow field and bed stress surrounding a short cylinder in response to combined wave and mean-flow forcing events is examined. Model simulations are performed with a 3-D nonhydrostatic computational fluid dynamics model, FLOW-3D. The model is forced with a range of characteristic tidal and wave velocities as observed in 12-15 m of water at the Martha's Vineyard Coastal Observatory (MVCO, Edgartown, MA). The 2.4-m-long and 0.5-m diameter cylinder is buried 10% of the diameter on a flat, fixed bed. Regions of incipient motion are identified through local estimates of the Shields parameter exceeding the critical value. Potential areas of sediment deposition are identified with local estimates of the Rouse parameter exceeding ten. The model predictions of sediment response are in general in agreement with field observations of seabed morphology obtained over a one-week period during the 2003-2004 MVCO mine burial experiment. Both observations and simulations show potential transport occurring at the ends of the mine in wave-dominated events. Mean flows greater than 10 cm/s lead to the formation of larger scour pits upstream of the cylinder. Deposition in both cases tends to occur along the sides, near the center of mass of the mine. However, the fixed-bed assumption prohibits the prediction of full perimeter scour as is observed in nature. Predicted scour and burial regimes for a range of wave and mean-flow combinations are established.

Modeling the nonlinear effect of wind on rectilinear tidal flow

Observations of near‐bed alongshore flow in 4–10 m depth at Terschelling, Netherlands, show that strong alongshore winds (≈15–20 m/s) in the flood direction reduce the tide current range and alter the tidal flow curve by bringing maximum flood flow forward in time and delaying maximum ebb flow. These nonlinear effects, which are more pronounced in shallower water, are reproduced well by a single‐point model in the vertical driven by large‐scale (O(km)) sea surface gradients and wind stress, using a quadratic bottom stress formulation and a time‐dependent eddy viscosity derived from a k‐ε turbulence closure model. Subsequent idealized model simulations using a (Terschelling based) M2 sea‐surface gradient and wind speeds between 0 and 20 m/s show that the wind‐induced modifications of the tidal flow are consistent with the interaction of the wind‐induced flow with the tidal flow through the quadratic bed stress. The wind‐driven flow enhances friction during the flood phase more than it reduces friction during the ebb flow, thereby increasing friction over a tidal cycle and, as a consequence, reducing the tide current range. The predicted increase in asymmetry of bed friction during flood and ebb flow with increasing wind speed also increases (decreases) the M4 (M6) amplitude, consistent with the observations. The M4/M2 amplitude ratio is predicted to be largest when the flow has just become unidirectional (that is, when the wind‐driven flow equals the tidal current).

Tidal evolution of the northwest European shelf seas from the Last Glacial Maximum to the present

Two‐dimensional paleotidal simulations have been undertaken to investigate tidal and tide‐dependent changes (tidal amplitudes, tidal current velocities, seasonal stratification, peak bed stress vectors) that have occurred in the NW European shelf seas during the last 20 ka. The simulations test the effect of shelf‐wide isostatic changes of sea level by incorporating results from two different crustal rebound models, and the effect of the ocean‐tide variability by setting open boundary values either fixed to the present state or variable according to the results of a global paleotidal model. The use of the different crustal rebound models does not affect the overall changes in tidal patterns, but the timing of the changes is sensitive to the local isostatic effects that differ between the models. The incorporation of ocean‐tide changes greatly augments the amplitude of tides and tidal currents in the Celtic and Malin seas before 10 ka BP, and has a large impact on the distribution of seasonally stratified conditions, magnitude of peak bed stress vectors and tidal dissipation in the shelf seas. The predictions on seasonal stratification are supported by well‐dated evidence on tidal mixing front migration in the Celtic Sea. Additional experiments using the global model suggest that the variability of offshore tides has been caused mainly by changes of eustatic sea level and ice‐sheet extent. In particular, a large decrease observed at 10–8 ka BP is attributed to the opening of Hudson Strait accompanied by the retreat of the Laurentide Ice Sheet.

Field evidence of pressure gradient induced incipient motion

Recently, models for the onshore migration of nearshore sandbars were improved by including an empirical acceleration term in the sediment transport formulation (Hoefel and Elgar, 2003). Here, field observations of the fluid‐sediment interface suggest that the success of this empirical parameter may result from unsteady forcing of sediment beds by nearshore waves. The velocity and sediment observations show that, under large pressure gradients present during the strengthening of onshore‐directed flow, several centimeters of previously still sediment can be mobilized and transported onshore. The onshore‐directed mobile‐bed layer rapidly settles as the wave velocity weakens. These observations are inconsistent with traditional sediment transport theories (Shields, 1936; Bagnold, 1966) that assume incipient motion is solely based on the force applied by bed stress but are consistent with theories that include an additional force induced by the horizontal pressure gradient (Madsen, 1974; Sleath, 1999). In this paper, we provide the first field evidence of incipient motion that is significantly influenced by the unsteady forcing introduced by the horizontal pressure gradient due to surface gravity waves.

A one-dimensional transient model of down-flow through a swelling packed porous bed

A transient model of down-flow through an ion-exchange column in which the resin swells has been developed. The model is herein described and results are presented. Wall friction can lead to high bed stresses when the resin in columns with high length to diameter ratios swells. These stresses can lead to high and potentially excursive hydraulic pressure drops along a column. A non-dimensional grouping that effectively correlates the final steady-state hydraulic behavior of a column, with the resin compressibility and column geometric and flow parameters, has been determined.

Exchange between a freshwater embayment and a large lake through a long, shallow channel

We analyze the exchange between a weakly forced lacustrine embayment and a large lake through a long, shallow channel. Exchange in the channel is the result of a multiple and subtle balance in which spatial thermal variations (baroclinic forcing), oscillations in the water level (barotropic forcing), bottom friction, diffusion, wind forcing, and the effects of unsteadiness are all important. Temperature gradients across the channel result from differences in thermal inertia of the embayment and lake at seasonal time scales and differences in the wind-driven internal dynamics of the lake and the embayment at diurnal to synoptic time scales. These gradients are, in general, weak and barotropic forcing dominates the channel momentum balance; however, episodic upwelling in the lake can shift the balance in favor of baroclinic dominance. A combination of scaling, analysis of observational data, and threedimensional simulations are used to demonstrate that bed stress, vertical turbulent diffusion, wind stress, and unsteadiness effect exchange relative to the predictions of internal hydraulic theoryߞthe quasisteady inviscid theory that describes the motion resulting from a purely barotropic/baroclinic force balance.

Analysis of 11 Bioengineered Stream Bank Erosion Control Structures in Alaska

This paper reports the results of a study of the use of bioengineered erosion control structures on Alaskan streams and rivers. Field investigations of hydraulic and vegetation conditions at 11 study sites around the state of Alaska were conducted to determine the performance of these structures. Root wads, live staking, brush layering, and coir logs were the primary bioengineering methods used for erosion control at the study sites. A one-dimensional numerical computer model was applied at each site to estimate the magnitude of average bed and bank shear stresses (tractive force) apparent to the erosion control structures at the 50- and 100-year design flood levels. Discharge records and field flood indicators were checked to correlate structure condition to flow history. Damage at existing structures was attributed to flowing ice, undermining of toe protection, buoyancy effects, and failure of construction fabrics. Root wad structures in good condition were located in areas with high boat wake occurrence but low channel tractive forces. The findings of the study suggest that the types of bioengineered erosion control structures studied have not been proven to offer reliable bank erosion protection during flooding conditions on channels with high tractive forces.

The effects of spatially complex inner shelf roughness on boundary layer turbulence and current and wave friction: Tairua embayment, New Zealand

A field study of bedforms, associated hydraulic roughness, and turbulence was conducted on the inner shelf off the east coast of New Zealand's North Island under conditions that included two significant storm events. Sharply contrasting rough and smooth beds were characterized via field mapping and deployment of instrumented benthic tripods. Rough areas of coarse sand exhibited ripples with heights and lengths of ∼25 and ∼100cm, while smooth areas supported smaller ripples with heights and lengths of ∼5cm and ∼20cm. Contacts between the two surfaces were sharp and maintained their position. Roughness contrasts were enhanced significantly during storms, which simultaneously accentuated migrating orbital ripples over the coarse bed and replaced ripples on the fine sediment bed with smoother hummocky features. Spectra of the fluctuating vertical velocity components, w′, from both smooth and rough sites showed good fits to −5/3 slopes within the inertial sub range enabling independent estimates of wave-averaged bed stress to be made via the inertial dissipation method (IDM). We also utilized the vertical fluctuation data to obtain alternative estimates of the wave friction factor, fw, following Smyth and Hay (J. Phys. Oceanography 32 (2002) 3490); SH. These two methods yielded generally similar results. Under high wave conditions, fw estimated via IDM averaged 0.027 at the rough site and 0.0045 at the smooth site while the SH method gave respective values of 0.027 and 0.013. Under low-energy conditions, fw from IDM averaged 0.0082 at the rough site and 0.012 at the smooth site, while the SH method yielded mean values of 0.0080 and 0.016. Thus, fw was much larger at the rough site than at the smooth site during storms but smaller at the rough site during fair weather. During storms, structured vortices with frequencies at the first harmonic of the swell waves formed over the rough surface and penetrated above the wave current boundary layer causing retardation of mean currents. Such storm-induced vortices were only intermittently present over the smooth surface. The application of Nielsen's (J. Geophys. Res. 86 (1981) 6467) roughness model produced some qualitatively similar trends in fw, although predicted fw was larger than observed values at the rough site. Additionally, low modelled fw over the smooth bed during high energy was based on plane bed theory rather than the inferred hummocky bed.

Modelling tidally induced sediment-transport paths over the northwest European shelf: the influence of sea-level reduction

A three-dimensional model covering the northwest European Shelf and part of the adjacent Atlantic Ocean is used to examine the influence of water depth change upon the distribution of maximum tidal bed stress. The direction of bed stress is an indicator of sediment movement as bed load and various regions of convergence and divergence in good agreement with observations are identified. Calculations are performed with water depths reduced by 35 m, corresponding to 10 000 years before present (B.P.). Initially, the model is forced by only the M2 tide, although subsequently five constituents, namely M2, S2, N2, K1 and O1, are used for tidal forcing. Although the distribution of extreme bed stresses computed with only M2 tidal forcing is comparable to that computed with five tides, the additional tidal constituents modify the magnitude of the bed stress. In particular the diurnal tides show regions of local enhanced current amplitude in the shelf-edge region with corresponding changes in bed stress. When water depths are reduced such that the North Sea and English Channel are separated, then there is a significant change in the tidal distribution in the shallow Southern Bight which influences bed-stress distributions and hence bed-load sediment transport in the area. Besides changes in shallow regions, the distribution of tides at the shelf edge is affected. A discussion of the limitations of the present coarse-grid model in shelf-edge regions and how it can be used to provide boundary conditions for limited-area three-dimensional models that can include stratification is presented. Also the importance of stratification for sediment movement at the shelf edge is briefly discussed.

Direct measurements of bed stress under swash in the field

Utilizing flush mounted hot film anemometry, the bed stress under swash was measured directly in a field experiment conducted on Barret Beach, Fire Island, New York. The theory, development, and calibration of the instrument package are discussed, and results from the field experiment are presented. Examples of bed stress time series throughout a swash cycle are presented, and an ensemble averaged swash bed stress cycle is calculated. Strong asymmetry is observed between the uprush and backwash phases of the swash flow. The maximum bed shear stress exerted by the uprush is approximately double that of the backwash, while the duration of the backwash is 135% greater than that of the uprush. Friction coefficients in the swash zone are observed to be similar in magnitude to those from steady flow, with the mean observed friction coefficient equal to 0.0037. Swash friction coefficients derived from the current measurements exhibit a Reynolds number dependence similar to that observed for other flows. A systematic difference between coefficients for uprush and backwash is suggested.