Effect of saltating sediment on flow resistance and bed roughness in overland flow

The acceleration of saltating grains by overland flow causes momentum to be transferred from the flow to the grains, thereby increasing flow resistance and bed roughness. To assess the impact of saltating sediment on overland flow hydraulics, velocity profiles in transitional and turbulent flows on a fixed sand-covered bed were measured using hot-film anemometry. Five discharges were studied. At each discharge, three flows were measured: one free of sediment, one with a relatively low sediment load, and one with a relatively high sediment load. In these flows from 83 to 90 per cent of the sediment was travelling by saltation. As a result, in the sediment-laden flows the near-bed velocities were smaller and the velocity profiles steeper than those in the equivalent sediment-free flows. Sediment loads ranged up to 87·0 per cent of transport capacity and accounted for as much as 20·8 per cent of flow resistance (measured by the friction factor) and 89·7 per cent of bed roughness (measured by the ratio of the roughness length to median grain diameter). It is concluded that saltating sediment has a considerable impact on overland flow hydraulics, at least on fixed granular beds. Saltation is likely to have a relatively smaller effect on overland flow on natural hillslopes and agricultural fields where form and wave resistance dominate. Still, saltation is generally of greater significance in overland flow than in river flow, and for this reason its effect on overland flow hydraulics is deserving of further study. © 1998 John Wiley & Sons, Ltd.     

Estimation of flow resistance in gravel-bedded rivers: A physical explanation of the multiplier of roughness length

The need to estimate velocity and discharge indirectly in gravel-bedded rivers is a commonly-encountered problem. Semilogarithmic friction equations are used to estimate mean velocity using a friction factor obtained from depth and grain size information. Although such equations have a semi-theoretical basis, in natural gravel-bed channels, an empirical constant (6.8 or 3.5) has to be introduced to scale-up the characteristic grain size (D₅₀ or D₈₄) to represent the effective roughness length. In this paper, two contrasting approaches are used to suggest that the multiplier of characteristic grain size is attributable to the effect of small-scale form resistance, reflecting the occurrence of microtopographic bedforms in gravel-bedded environments. First, spatial elevation dependence in short, detailed bed profiles from a single gravel-bedded river is investigated using semivariogram and zero-crossing analyses. This leads to objective identification of two discrete scales of bed roughness, associated with grain and microtopographic roughness elements. Second, the autocorrelation structure of the three-dimensional near-bed velocity field is examined to identify regularities associated with eddy shedding and energy losses from larger grains and microtopographic bedforms. Apart from improving the capacity to determine friction factors for velocity and discharge estimation, the findings have implications in general for the initial motion of gravelly bed material.     

Experimental study on the effects of artificial bed roughness on turbidity currents over abrupt bed slope change

A series of experimental observations are presented in the current study to discuss the effects of artificial bed roughness on the turbidity current flowing in a rectangular channel with an abrupt change in bed slope.For this purpose,two different types of elements,sinusoidal and trapezoidal,with various heights and arrangements are considered as artificial bed roughness.A Vectrino velocity meter was used to measure the velocity and sediment concentration profiles.The effects of inlet sediment concentration on front velocity,body velocity,unit discharge,sediment concentration,and suspended load transport rate also were investigated.Accurate equations were developed for estimation of the velocity of a turbidity current over smooth and rough beds.The unexpected experimental results showed that unlike the effect of roughness height,a change in the roughness arrangement has no significant influence on the velocity of a turbidity current.Also,the effect of bed roughness on the front velocity of a denser current is more significant.     

3D float tracking: in situ floodplain roughness estimation

This paper presents a novel technique to quantify in situ hydrodynamic roughness of submerged floodplain vegetation: 3D float tracking. This method uses a custom‐built floating tripod that is released on the inundated floodplain and tracked from shore by a robotic total station. Simultaneously, an acoustic Doppler current profiler (ADCP) collects flow velocity profiles and water depth data. Roughness values are derived from two methods based on (1) run‐averaged values of water depth, slope and flow velocity to compute the roughness based on the Chézy equation, assuming uniform flow, (2) the equation for one‐dimensional free surface flow in a moving window. A sensitivity analysis using synthetic data proved that the median value of the roughness, derived using method 2, is independent of (1) the noise in water levels, up to 9 mm, (2) bottom surface slope, and (3) topographic undulations. The window size should be at least 40 m for a typical lowland river setup. Field measurements were carried out on two floodplain sections with an average vegetation height of 0·030 (Arnhem) and 0·043 m (Dreumel). Method 1 resulted in a Nikuradse roughness length of 0·08 m for both locations. Method 2 gave 0·12 m for Arnhem and 0·19 m for Dreumel. In Arnhem, a spatial pattern of roughness values was present, which might be related to fractional vegetation cover or vegetation density during the flood peak. 3D float tracking proved a flexible and detailed method for roughness determination in the absence of waves, and provided an unrestricted view from shore. Copyright © 2008 John Wiley & Sons, Ltd.     

The influence of pool length on local turbulence production and energy slope: a flume experiment

The in?uence of pool length on the strength of turbulence generated by vortex shedding was investigated in a 6 m long recirculating ?ume. The experiment utilized a 38% constriction of ?ow and an average channel‐bed slope of 0·007. The base geometry for the intermediate‐length pool experiment originated from a highly simpli?ed, 0·10 scale model of a forced pool from North Saint Vrain Creek, Colorado. Discharge in the ?ume was 31·6 l/s, which corresponds to a discharge in the prototype channel of 10 m³/s. Three shorter and four longer pool lengths also were created with a ?xed bed to determine changes in turbulence intensities and energy slope with pool elongation. Three‐dimensional velocities were measured with an acoustic Doppler velocimeter at 31–40 different 0·6‐depth and near‐bed locations downstream of the rectangular constriction. The average velocity and root mean square (RMS) of the absolute magnitude of velocity at both depths are signi?cantly related to the distance from the constriction in most pool locations downstream of the constriction. In many locations, pool elongation results in a non‐linear change in turbulence intensities and average velocity. Based on the overall ?ow pattern, the strongest turbulence occurs in the center of the pool along the shear zone between the jet and recirculating eddy. The lateral location of this shear zone is sensitive to changes in pool length. Energy slope also was sensitive to pool length due to a combination of greater length of the pool and greater head loss with shorter pools. The results indicate some form of hydraulic optimization is possible with pools adjusting their length to adjust the location and strength of turbulent intensities in the center of pools, and lower their rate of energy dissipation. Copyright © 2004 John Wiley & Sons, Ltd.     

Sensitivity of bed shear stress estimated from vertical velocity profiles: the problem of sampling resolution

Bed shear stress in open channel flows is often estimated from the logarithmic vertical velocity profile. However, most measuring devices used in the field do not allow for flow velocity to be measured very close to the bed. The lack of near-bed measurements is a critical loss of information which may affect bed shear stress estimates. Detailed velocity profiles obtained from a field acoustic Doppler velocimeter over three different bed roughnesses clearly show that the inclusion of near-bed points is critical for the estimation of bed shear stress in a shallow river environment. Moreover, the results indicate that using the full flow depth instead of the bottom 20 per cent of the profile generates an underestimation of the shear stress when flow is uniform. © 1998 John Wiley & Sons, Ltd.     

Study on effect of surface roughness on overland flow from different geometric surfaces through numerical simulation

Effect of variability in surface roughness on overland flow from different geometric surfaces is investigated using numerical solution of diffusion wave equation. Three geometric surfaces rectangular plane, converging and diverging plane at slopes 1 to 3% are used. Overland flow is generated by applying rainfall at constant intensity of 10 mm/h for period 30 min and 100 min. Three scenarios of spatial roughness conditions viz. roughness increasing in downstream direction, roughness decreasing in downstream direction and roughness distributed at random are considered. Effect of variability of roughness on overland flow in terms of depth, velocity of flow and discharge along the distance from upstream to downstream for different geometric surfaces are discussed in detail. Results from the study indicate that roughness distribution has significant effect on peak, time to peak and overall shape of the overland flow hydrograph. The peak occurs earlier for the scenario when roughness increases in downstream direction as compared to scenario when roughness is decreasing in downstream for all three geometric surfaces due to very low friction factor and more velocity at the top of the domain. The converging plane attains equilibrium state early as compared to rectangular and diverging plane. Different set of random values result in different time to peak and shape of hydrograph for rectangular and diverging plane. However, in case of converging plane, the shape of computed hydrographs remains almost similar for different sets of random roughness values indicating stronger influence of converging geometry than effect due to variation of roughness sequence on computed runoff hydrograph. Hierarchically, the influence of geometry on overland flow is stronger than the influence of slope and the influence of slope is stronger than the influence of roughness. Copyright © 2013 John Wiley & Sons, Ltd.     

Aeolian sediment flux decay: Non-linear behaviour on developing deflation lag surfaces

Wind tunnel simulations of the effect of non-erodible roughness elements on sediment transport show that the flux ratio q/q_s, shear velocity U*, and roughness density λ are co-dependent variables. Initially, the sediment flux is enhanced by kinetic energy retention in relatively elastic collisions that occur at the roughness element surfaces, but at the same time, the rising surface coverage of the immobile elements reduces the probability of grain ejection. A zone of strong shearing stress develops within 0·03 to 0·04 m of the rough bed because of a relative straightening of velocity profiles which are normally convex with saltation drag. This positive influence on fluid entrainment is opposed by declining shear stress partitioned to the sand bed. Similarly, because the free stream velocity U_f is fixed while U_* increases, velocity at height z and particle momentum gain from the airstream decline, leading eventually to lower numbers of particles ejected on average at each impact. When the ratio of the element basal area to frontal area σ is approximately equal to 3·5, secondary flow effects appear to become significant, so that the dimensionless aerodynamic roughness parameter Z₀/h and shear stress on the exposed sand bed T_s decrease. It is at this point that grain supply to the airstream and saltation drag appear to be significantly reduced, thereby intensifying the reduction in U_*. The zone of strong fluid shear near the bed dissipates.     

Flow resistance and hydraulic geometry in bedrock rivers with multiple roughness length scales

Many models of incision by bedrock rivers predict water depth and shear stress from discharge; conversely, palaeoflood discharge is sometimes reconstructed from flow depth markers in rock gorges. In both cases, assumptions are made about flow resistance. The depth–discharge relation in a bedrock river must depend on at least two roughness length scales (exposed rock and sediment cover) and possibly a third (sidewalls). A conceptually attractive way to model the depth–discharge relation in such situations is to partition the total shear stress and friction factor, but it is not obvious how to quantify the friction factor for rough walls in a way that can be used in incision process models. We show that a single flow resistance calculation using a spatially averaged roughness length scale closely approximates the partitioning of stress between sediment and rock, and between bed and walls, in idealized scenarios. Both approaches give closer fits to the measured depth–discharge relations in two small bedrock reaches than can be achieved using a fixed value of Manning's n or the Chézy friction factor. Sidewalls that are substantially rougher or smoother than the bed have a significant effect on the partitioning of shear stress between bed and sidewalls. More research is needed on how best to estimate roughness length scales from observable or measurable channel characteristics. © 2019 John Wiley & Sons, Ltd.     

Velocity characteristics along a small step–pool channel

This paper summarizes measurements of velocity along three reaches of a small mountain channel with step–pool bedforms. A one‐dimensional electromagnetic current meter was used to record velocity fluctuations at 37 fixed measurement points during five measurement intervals spanning the peak of the annual snowmelt hydrograph. Measurement cross‐sections were located upstream from a bed‐step, at the step lip, downstream from the step, and in a uniform‐gradient run. Data analyses focused on characteristics of velocity profiles, and on correlations between velocity characteristics and the potential control variables bedform type, reach gradient and flow depth. To test the hypothesis that velocity characteristics are related to channel bedform types, ANOVA and ANCOVA tests were performed for the average velocity and coefficient of variation of point velocity data. Results indicate that high frequency velocity variations correlate to some degree with both channel characteristics and discharge. Velocity became more variable as stage increased, particularly at low‐gradient reaches with less variable bed roughness. Velocity profiles suggest that locations immediately downstream from bed‐steps are dominated by wake turbulence from mid‐profile shear layers. Locations immediately upstream from steps, at step lips, and in runs are dominated by bed‐generated turbulence. Adverse pressure gradients upstream and downstream from steps may be enhancing turbulence generation, whereas favourable pressure gradients at steps are suppressing turbulence. The bed‐generated turbulence and skin friction of runs appear to be less effective energy dissipators than the wake‐generated turbulence and form drag of step–pool bedforms. Copyright © 2000 John Wiley & Sons, Ltd.     

Rill hydraulics on a semiarid hillslope,southern Arizona

Seventy field experiments were conducted in seven rills located on a semiarid rangeland hillslope underlain by gravelly soils at Walnut Gulch, Arizona. The rills, which are characterized by wide, shallow cross-sections and gravel-covered beds, have mean at-a-station hydraulic geometry exponents of b = 0·33, f = 0·34 and m = 0·33. Although the differences between these values and typical values of b = 0·30, f = 0·40 and m = 0·30 for cropland rills are not statistically significant, they are thought to be real, as cropland rills often have more rectangular cross-sections and steeper sides than the rangeland rills under study. For rills formed in silty loamy soils, Govers developed an empirical relation between mean flow velocity and discharge. Emphasizing the generality of this relation, he suggested that it may be used as a simple means of routing runoff through rills. He also noted that this relation appeared to be unaffected by either slope or soil materials. The present data represent rills underlain by coarser and somewhat more varied gravel-rich soils. These data do not conform to Govers' relation, and a multiple regression analysis reveals that slope and soil materials, either directly or indirectly through bed roughness, exert almost as much influence on flow velocity as does discharge. Three alternative methods are developed for predicting flow velocity in the rills under study. All three methods give good results with the largest root mean square deviation being 3·115 cm s⁻¹.     

Roughness coefficients and velocity estimation in well-inundated sheet and rilled overland flow without strongly eroding bed forms

Even with the flow of water over a soil surface in which roughness elements are well inundated, and in less erosive situations where erosional bed forms are not pronounced, the magnitude of resistance coefficients in equations such as those of Darcy–Weisbach, Chezy or Manning vary with flow velocity (at least). Using both original laboratory and field data, and data from the literature, the paper examines this question of the apparent variation of resistance coefficients in relation to flow velocity, even in the absence of interaction between hydraulics and resulting erosional bed forms. Resistance equations are first assessed as to their ability to describe overland flow velocity when tested against these data sources. The result is that Manning's equation received stronger support than the Darcy–Weisbach or Chezy equations, though all equations were useful. The second question addressed is how best to estimate velocity of overland flow from measurements of slope and unit discharge, recognizing that the apparent flow velocity variation in resistance coefficients is probably a result of shortcomings in all of the listed resistance equations. A new methodology is illustrated which gives good agreement between estimated and measured flow velocity for both well-inundated sheet and rill flow. Comments are given on the predictive use of this methodology. Copyright © 1999 John Wiley & Sons, Ltd.     

An experimental investigation of the effects of thawed soil depth on rill flow velocity

Water flow velocity is an important hydraulic variable in hydrological and soil erosion models, and is greatly affected by freezing and thawing of the surface soil layer in cold high-altitude regions. The accurate measurement of rill flow velocity when impacted by the thawing process is critical to simulate runoff and sediment transport processes. In this study, an electrolyte tracer modelling method was used to measure rill flow velocity along a meadow soil slope at different thaw depths under simulated rainfall. Rill flow velocity was measured using four thawed soil depths (0, 1, 2 and 10 cm), four slope gradients (5°, 10°, 15° and 20°) and four rainfall intensities (30, 60, 90 and 120 mm·h⁻¹). The results showed that the increase in thawed soil depth caused a decrease in rill flow velocity, whereby the rate of this decrease was also diminishing. Whilst the rill flow velocity was positively correlated with slope gradient and rainfall intensity, the response of rill flow velocity to these influencing factors varied with thawed soil depth. The mechanism by which thawed soil depth influenced rill flow velocity was attributed to the consumption of runoff energy, slope surface roughness, and the headcut effect. Rill flow velocity was modelled by thawed soil depth, slope gradient and rainfall intensity using an empirical function. This function predicted values that were in good agreement with the measured data. These results provide the foundation for a better understanding of the effect of thawed soil depth on slope hydrology, erosion and the parameterization scheme for hydrological and soil erosion models.     

An experimental study on the evolution law of roll wave parameters in overland flow

Roll waves commonly occur in overland flow and have an important influence on the progress of soil erosion on slopes. This study aimed to explore the evolution and mechanism of roll waves on steep slopes. The potential effects of flow rate, rainfall intensity and bed roughness on the laws controlling roll wave parameters were investigated. The flow rates, rainfall intensities and bed roughness varied from 5 to 30 L/min, 0 to 150 mm/h, and 0.061 to 1.700 mm, respectively. The results indicate that roll waves polymerize significantly along the propagation path, and bed roughness and rainfall affect the generation and evolution of roll waves. The wave velocity, length and height decreased with bed roughness, whereas the wave frequency increased with increasing bed roughness under fixed flow rate and rainfall intensity conditions. Rainfall increased the wave velocity and wavelength and decreased the wave frequency. The wave velocity, height and wavelength tended to increase with an increasing flow rate. Rainfall promoted the generation of roll waves, whereas bed roughness had the opposite effect. The generation of roll waves is closely related to the Froude number (Fr) and flow resistance. In this experiment, the range of the Reynolds number for the roll waves generated in the laminar region was 142–416, and the range of the flow resistance coefficient was 0.64–4.85. The critical value of the Fr for flow instability in the laminar region was approximately 0.57. Exploring the generation and evolution law of roll waves is necessary for understanding the processes and dynamic mechanisms of slope soil erosion.     

Bed roughness and grain sorting-an experimental study over fine to medium sand beds

The relation between grain-size distribution of the bed and in suspension was critically examined under a uniform flow velocity of 50 cm/s over two beds: one of mainly fine sands and the other of medium sands. Two sections – one 2.85 m downstream and the other 6.35 m downstream in the experimental channel-were selected for sampling to study the grain-sorting pattern in the vertical direction along the direction of transport. The shape and type of the grain-size distribution pattern were critically studied with height above the bed. The change in the distribution pattern has been attributed to the change of local bed roughness causing scouring against the protruded relatively coarse grains on the bed. Such trends are important to predict the nature of river bed topography. The sand of Bed-1 initially exhibits a log-skew-Laplace distribution at different heights of suspension. The distribution pattern, however, changes but this changing pattern is not consistent along the upstream side. For Bed-2, which initially exhibits a log-normal distribution, the same pattern persists from the height of suspension at 5 cm up to 20 cm. Such consistency in log-normality is also observed at the downstream points of measurement. It is generally expected that the mean grain-size would reduce with increases of suspension height but the results of the experiments, in some occasions, differ significantly from the gradual fining upward trend. This result has been attributed to local changes of bed roughness arising from the protruded relatively coarse grains causing eddies, scouring, and turbulent phenomena which moves coarse particles higher in suspension adding a coarse tail to the distribution increasing the mean grain-size.     

Computational fluid dynamics modelling of boundary roughness in gravel‐bed rivers: an investigation of the effects of random variability in bed elevation

Results from a series of numerical simulations of two‐dimensional open‐channel flow, conducted using the computational fluid dynamics (CFD) code FLUENT, are compared with data quantifying the mean and turbulent characteristics of open‐channel flow over two contrasting gravel beds. Boundary roughness effects are represented using both the conventional wall function approach and a random elevation model that simulates the effects of supra‐grid‐scale roughness elements (e.g. particle clusters and small bedforms). Results obtained using the random elevation model are characterized by a peak in turbulent kinetic energy located well above the bed (typically at y/h = 0·1–0·3). This is consistent with the field data and in contrast to the results obtained using the wall function approach for which maximum turbulent kinetic energy levels occur at the bed. Use of the random elevation model to represent supra‐grid‐scale roughness also allows a reduction in the height of the near‐bed mesh cell and therefore offers some potential to overcome problems experienced by the wall function approach in flows characterized by high relative roughness. Despite these benefits, the results of simulations conducted using the random elevation model are sensitive to the horizontal and vertical mesh resolution. Increasing the horizontal mesh resolution results in an increase in the near‐bed velocity gradient and turbulent kinetic energy, effectively roughening the bed. Varying the vertical resolution of the mesh has little effect on simulated mean velocity profiles, but results in substantial changes to the shape of the turbulent kinetic energy profile. These findings have significant implications for the application of CFD within natural gravel‐bed channels, particularly with regard to issues of topographic data collection, roughness parameterization and the derivation of mesh‐independent solutions. Copyright © 2001 John Wiley & Sons, Ltd.     

Comparing flow resistance law for fixed and mobile bed rills

Rills caused by run‐off concentration on erodible hillslopes have very irregular profiles and cross‐section shapes. Rill erosion directly depends on the hydraulics of flow in the rills, which may differ greatly from hydraulics of flow in larger and regular channels. In this paper, a recently theoretically deduced rill flow resistance equation, based on a power–velocity profile, was tested experimentally on plots of varying slopes (ranging from 9% to 26%) in which mobile and fixed bed rills were incised. Initially, measurements of flow velocity, water depth, cross‐section area, wetted perimeter, and bed slope, carried out in 320 reaches of mobile bed rills and in 165 reaches of fixed rills, were used for calibrating the theoretical flow resistance equation. Then the relationship between the velocity profile parameter Γ, the channel slope, and the flow Froude number was separately calibrated for the mobile bed rills and for the fixed ones. The measurements carried out in both conditions (fixed and mobile bed rills) confirmed that the Darcy–Weisbach friction factor can be accurately estimated using the proposed theoretical approach. For mobile bed rills, the data were supportive of the slope independence hypothesis of velocity, due to the feedback mechanism, stated by Govers. The feedback mechanism was able to produce quasicritical flow conditions. For fixed bed rills, obtained by fixing the rill channel, by a glue, at the end of the experimental run with a mobile bed rill, the slope independence of the flow velocity measurements was also detected. Therefore, an experimental run carried out by a rill bed fixed after modelling flow action is useful to detect the feedback mechanism. Finally, the analysis showed that, for the investigated conditions, the effect of sediment transport on the flow resistance law can be considered negligible respect to the grain roughness effect.     

Effect of bed surface roughness on longitudinal dispersion in artificial open channels

In several empirical and modelling studies on river hydraulics, dispersion was negatively correlated to surface roughness. In this study, it was aimed to investigate the influence of surface roughness on longitudinal dispersion under controlled conditions. In artificial flow channels with a length of 104 m, tracer experiments with variations in channel bed material were performed. By use of measured tracer breakthrough curves, average flow velocity, mean longitudinal dispersion, and mean longitudinal dispersivity were calculated. Longitudinal dispersion coefficients ranged from 0·018 m² s^?1 in channels with smooth bed surface up to 0·209 m² s^?1 in channels with coarse gravel as bed material. Longitudinal dispersion was linearly related to mean flow velocity. Accordingly, longitudinal dispersivities ranged between 0·152 ± 0·017 m in channels with smooth bed surface and 0·584 ± 0·015 m in identical channels with a coarse gravel substrate. Grain size and surface roughness of the channel bed were found to correlate positively to longitudinal dispersion. This finding contradicts several existing relations between surface roughness and dispersion. Future studies should include further variation in surface roughness to derive a better‐founded empirical equation forecasting longitudinal dispersion from surface roughness. Copyright © 2011 John Wiley & Sons, Ltd.     

SCOUR HOLE CHARACTERISTICS BELOW FREE OVERFALL SPILLWAY

1 INTRODUCTION Flow through hydraulic structures often issues in the form of jets. The jet velocities are usually high enough to produce sizable, even dangerous scour hole. The extent of the resulting scour depends on the nature of bed material and flow characteristics. The erosion process is quite complex and depends upon the interaction of hydraulic and morphological factors. Scouring may lead to: endangering the stability of the structure by structural failure or increased seepage, end…     

Parameterization of near-bed processes under collinear wave and current flows from a two-phase sheet flow model

Sediment transport models require appropriate representation of near-bed processes. We aim here to explore the parameterizations of bed shear stress, bed load transport rate and near-bed sediment erosion rate under the sheet flow regime. To that end, we employ a one-dimensional two-phase sheet flow model which is able to resolve the intrawave boundary layer and sediment dynamics at a length scale on the order of the sediment grain. We have conducted 79 numerical simulations to cover a range of collinear wave and current conditions and sediment diameters in the range 210–460 μm$\mathrm{\mu }\mathrm{m}$. The numerical results confirm that the intrawave bed shear stress leads the free stream velocity, and we assess an explicit expression relating the phase lead to the maximum velocity, wave period and bed roughness. The numerical sheet flow model is also used to provide estimates for the bed load transport rate and to inspect the near-bed sediment erosion. A common bed load transport rate formulation and two typical reference concentration approaches are assessed. A dependence of the bed load transport rate on the sediment grain diameter is observed and parameterized. Finally, the intrawave near-bed vertical sediment flux is further investigated and related to the time derivative of the bed shear stress.