Near‐bed shear stress,turbulence production and dissipation in a shallow and narrow tidal channel

Near‐bed, highly resolved velocity profiles were measured in the lower 0.03 m of the water column using acoustic Doppler profiling velocimeters in narrow tidal channels in a salt marsh. The bed shear stress was estimated from the velocity profiles using three methods: the log‐law, Reynolds stress, and shear stress derived from the turbulent kinetic energy (TKE). Bed shear stresses were largest during ebbing tide, while near‐bed velocities were larger during flooding tide. The Reynolds stress and TKE method gave similar results, while the log‐law method resulted in smaller bed shear stress values during ebbing tide. Shear stresses and turbulent kinetic energy followed a similar trend with the largest peaks during ebbing tide. The maximum turbulent kinetic energy was on the order of 1 × 10^{? 2} m²/s². The fluid shear stress during flooding tide was approximately 30% of the fluid shear stress during ebbing tide. The maximum TKE‐derived shear stress was 0.7 N/m² and 2.7 N/m² during flooding and ebbing tide, respectively, and occurred around 0.02 m above the bed. Turbulence dissipation was estimated using the frequency spectrum and structure function methods. Turbulence dissipation estimates from both methods were maximum near the bed (~0.01 m). Both the structure function and the frequency spectrum methods resulted in maximum dissipation estimates on the order of 4 × 10^{? 3} m²/s³. Turbulence production exceeded turbulence dissipation at every phase of the tide, suggesting that advection and vertical diffusion are not negligible. However, turbulence production and dissipation were within a factor of 2 for 77% of the estimates. The turbulence production and dissipation decreased quickly away from the bed, suggesting that measurements higher in the water column cannot be translated directly to turbulence production and dissipation estimates near the bed. Copyright © 2015 John Wiley & Sons, Ltd.     

Effect of non‐uniformity of flow on velocity and turbulence intensities over a cobble‐bed

Non‐uniform flows encompassing both accelerating and decelerating flows over a cobble‐bed flume have been experimentally investigated in a flume at a scale of intermediate relative submergence. Measurements of mean longitudinal flow velocity u, and determinations of turbulence intensities u′, v′, w′, and Reynolds shear stress ?u_fw_f have been made. The longitudinal velocity distribution was divided into the inner zone close to the bed and the outer zone far from the bed. In the inner zone of the boundary layer (near the bed) the velocity profile closely followed the ‘Log Law’; however, in the outer zone the velocity distribution deviated from the Log Law consistently for both accelerating and decelerating flows and the changes in bed slopes ranging from ?2% to + 2% had no considerable effect on the outer zone. For a constant bed slope (S = ±0·015), the larger the flow rate, the smaller the turbulence intensities. However, no detectable pattern has been observed for u′, v′ and w′ distributions near the bed. Likewise, for a constant flow rate (Q = 0·040 m³/s), with variation in bed slope the longitudinal turbulent intensity profile in the longitudinal direction remained concave for both accelerating and decelerating flows; whereas vertical turbulent intensity (w′) profile presented no specific form. The results reveal that the positions of maximum values of turbulence intensities and the Reynolds shear stress depend not only on the flow structure (accelerating or decelerating) but also on the intermediate relative submergence scale. Copyright © 2009 John Wiley & Sons, Ltd.     

The influence of riparian vegetation on near‐bank turbulence: a flume experiment

Measurements from a fixed‐bed, Froude‐scaled hydraulic model of a stream in northeastern Vermont demonstrate the importance of forested riparian vegetation effects on near‐bank turbulence during overbank flows. Sections of the prototype stream, a tributary to Sleepers River, have increased in channel width within the last 40 years in response to passive reforestation of its riparian zone. Previous research found that reaches of small streams with forested riparian zones are commonly wider than adjacent reaches with non‐forested, or grassy, vegetation; however, driving mechanisms for this morphologic difference are not fully explained. Flume experiments were performed with a 1:5 scale, simplified model of half a channel and its floodplain, mimicking the typical non‐forested channel size. Two types of riparian vegetation were placed on the constructed floodplain: non‐forested, with synthetic grass carpeting; and forested, where rigid, randomly distributed, wooden dowels were added. Three‐dimensional velocities were measured with an acoustic Doppler velocimeter at 41 locations within the channel and floodplain at near‐bed and 0·6‐depth elevations. Observations of velocity components and calculations of turbulent kinetic energy (TKE), Reynolds shear stress and boundary shear stress showed significant differences between forested and non‐forested runs. Generally, forested runs exhibited a narrow band of high turbulence between the floodplain and main channel, where TKE was roughly two times greater than TKE in non‐forested runs. Compared to non‐forested runs, the hydraulic characteristics of forested runs appear to create an environment with higher erosion potential. Given that sediment entrainment and transport can be amplified in flows with high turbulence intensity and given that mature forested stream reaches are wider than comparable non‐forested reaches, our results demonstrated a possible driving mechanism for channel widening during overbank flow events in stream reaches with recently reforested riparian zones. Copyright © 2007 John Wiley & Sons, Ltd.     

Comparing different methods of bed shear stress estimates in simple and complex flow fields

Bed shear stress is a fundamental variable in river studies to link ?ow conditions to sediment transport. It is, however, dif?cult to estimate this variable accurately, particularly in complex ?ow ?elds. This study compares shear stress estimated from the log pro?le, drag, Reynolds and turbulent kinetic energy (TKE) approaches in a laboratory ?ume in a simple boundary layer, over plexiglas and over sand, and in a complex ?ow ?eld around de?ectors. Results show that in a simple boundary layer, the log pro?le estimate is always the highest. Over plexiglas, the TKE estimate was the second largest with a value 30 per cent less than the log estimate. However, over sand, the TKE estimate did not show the expected increase in shear stress. In a simple boundary layer, the Reynolds shear stress seems the most appropriate method, particularly the extrapolated value at the bed obtained from a turbulent pro?le. In a complex ?ow ?eld around de?ectors, the TKE method provided the best estimate of shear stress as it is not affected by local streamline variations and it takes into account the increased streamwise turbulent ?uctuations close to the de?ectors. It is suggested that when single‐point measurements are used to estimate shear stress, the instrument should be positioned close to 0·1 of the ?ow depth, which corresponds to the peak value height in pro?les of Reynolds and TKE shear stress. Copyright © 2004 John Wiley & Sons, Ltd.     

Determination of critical shear stress of non‐cohesive soils using submerged jet test and turbulent kinetic energy

Laboratory tests using Jet Erosion Testing (JET) apparatus, impinging normally on a horizontal boundary, were conducted to determine the critical shear stress (τ_c) of non‐cohesive soil samples. A three‐dimensional (3D) SonTek/YSI 16 MHz Micro‐Acoustic Doppler Velocimeter (MicroADV) was used to measure turbulent kinetic energy (TKE) at a radial limit of entrainment in the wall jet zone and the measurements were used to calculate τ_c of the samples. The results showed that TKE increases exponentially with increasing particle size. The τ_c from this study were comparable (R² = 0.8) to the theoretical τ_c from Shields diagram after bed roughness scale ratio (D/k_s), due to the non‐uniform bed conditions, was accounted for. This study demonstrated that JET and TKE can be used to determine τ_c of non‐cohesive soils. The use of JET and TKE was found to be faster and easier when compared to the conventional approach of using flumes. A relationship of TKE at the onset of incipient motion (TKE_c) and samples’ D₅₀ developed in this study can be used to predict τ_c of non‐cohesive soils under similar non‐uniform conditions. Copyright © 2015 John Wiley & Sons, Ltd.     

Experimental investigation of the impacts of coherent flow structures upon turbulence properties in regions of crescentic scour

This study examines the spatial distributions of third‐order moments of velocity fluctuations, the turbulent kinetic energy (TKE) fluxes, and the conditional statistics of Reynolds shear stress across the equilibrium crescentic scour structures generated upstream of short horizontal static cylinders. Detailed velocity data were collected using three‐dimensional (3D) micro‐acoustic Doppler velocimeter (ADV) across and within the equilibrium scour marks. The analysis reveals that the positive and negative values of third‐order moments associated with the level bed surface and the scour holes are directly related to coherent structures. The components of TKE flux are discussed for the near‐bed region of the level bed surface and scour holes in relation to sweep–ejection events. A cumulant‐discard method is applied to the Gram‐Charlier probability distribution of two variables to describe the statistical properties of the term u′w′. The conditional statistics of the Reynolds shear stress show a good agreement with the experimental data. The distribution of the joint probability density function in the near‐bed region changes cyclically along the scour hole depending on the bottom fluid velocity, which implies a change from upward to downward flux of momentum and vice versa. Both the ejection and sweep events at near‐bed points on the level surface are more important than within the scour region; and in contrast, both events are stronger for the scour marks than the level bed surface at the outer layer. Sweeps dominate over ejections for the scour hole induced by smaller diameter and ejections dominate for larger diameter. Copyright © 2013 John Wiley & Sons, Ltd.     

A bed‐load transport model for rough turbulent open‐channel flows on plane beds

Data from flume studies are used to develop a model for predicting bed‐load transport rates in rough turbulent two‐dimensional open‐channel flows moving well sorted non‐cohesive sediments over plane mobile beds. The object is not to predict transport rates in natural channel flows but rather to provide a standard against which measured bed‐load transport rates influenced by factors such as bed forms, bed armouring, or limited sediment availability may be compared in order to assess the impact of these factors on bed‐load transport rates. The model is based on a revised version of Bagnold's basic energy equation i_bs_b = e_bω, where i_b is the immersed bed‐load transport rate, ω is flow power per unit area, e_b is the efficiency coefficient, and s_b is the stress coefficient defined as the ratio of the tangential bed shear stress caused by grain collisions and fluid drag to the immersed weight of the bed load. Expressions are developed for s_b and e_b in terms of G, a normalized measure of sediment transport stage, and these expressions are substituted into the revised energy equation to obtain the bed‐load transport equation i_b = ω G ^3·4. This equation applies regardless of the mode of bed‐load transport (i.e. saltation or sheet flow) and reduces to i_b = ω where G approaches 1 in the sheet‐flow regime. That i_b = ω does not mean that all the available power is dissipated in transporting the bed load. Rather, it reflects the fact that i_b is a transport rate that must be multiplied by s_b to become a work rate before it can be compared with ω. It follows that the proportion of ω that is dissipated in the transport of bed load is i_bs_b/ω, which is approximately 0·6 when i_b = ω. It is suggested that this remarkably high transport efficiency is achieved in sheet flow (1) because the ratio of grain‐to‐grain to grain‐to‐bed collisions increases with bed shear stress, and (2) because on average much more momentum is lost in a grain‐to‐bed collision than in a grain‐to‐grain one. Copyright © 2006 John Wiley & Sons, Ltd.     

Variable and conflicting shear stress estimates inside a boundary layer with sediment transport

This paper presents a comparison between two methods for estimating shear stress in an atmospheric internal boundary layer over a beach surface under optimum conditions, using wind velocities measured synchronously at 13 heights over a 1.7 m vertical array using ultrasonic anemometry. The Reynolds decomposition technique determines at‐a‐point shear stresses at each measurement height, while the Law‐of‐the‐Wall yields a single boundary layer estimate based on fitting a logarithmic velocity profile through the array data. Analysis reveals significant inconsistencies between estimates derived from the two methods, on both a whole‐event basis and as time‐series. Despite a near‐perfect fit of the Law‐of‐the‐Wall, the point estimates of Reynolds shear stress vary greatly between heights, calling into question the assumed presence of a constant stress layer. A comparison with simultaneously measured sediment transport finds no relationship between transport activity and the discrepancies in shear stress estimates. Results do show, however, that Reynolds shear stress measured nearer the bed exhibits slightly better correlation with sand transport rate. The findings serve as a major cautionary message to the interpretation and application of single‐height measurements of Reynolds shear stress and their equivalence to Law‐of‐the‐Wall derived estimates, and these concerns apply widely to boundary layer flows in general. © 2015 The Authors. Earth Surface Processes and Landforms published by John Wiley & Sons Ltd.     

Bank‐toe processes in incised channels: the role of apparent cohesion in the entrainment of failed bank materials

Numerous processes may instigate bank retreat and the consequent collection of failed cohesive materials at the bank toe. Cohesion between the failed material and the substrate can provide additional strength to resist direct fluvial entrainment. Failed, cohesive material can act as a form of natural bank‐toe protection by consuming and diverting flow energy that may otherwise be used to further scour the basal zone of incising channels. Investigations in Goodwin Creek, Mississippi, have revealed the existence of apparent cohesion between failed, cohesive blocks and their underlying surface. The method used to assess this cohesion involved a pulley system mounted on a tripod and supporting a load cell. Mean and maximum apparent‐cohesion values of 1·08 kPa and 2·65 kPa, respectively, were measured in this way, identifying a source that bonds blocks to the underlying surface. Cohesion values and types vary spatially and temporally. Tensiometric tests beneath blocks suggest that cohesion resulting from matric suction alone may be as much as 3·5 kPa in summer and 1·8 kPa in winter. Apparent cohesion is believed to have been sufficient to help prevent removal of the largest blocks by a peak flow of 66·4m³/s on 23 September 1997. Maximum excess shear stress required to entrain a D₇₅ block can be augmented by as much as 97% by the presence of apparent cohesion at the block–substrate interface when compared with a condition with zero apparent cohesion at the block underside. Given these findings, it is no longer sufficient to estimate block entrainment in the basal area from block size or bed roughness alone, as in a Shields‐type approach. Copyright © 2001 John Wiley & Sons, Ltd.     

A field study of mean and turbulent flow characteristics upwind,over and downwind of barchan dunes

Field‐measured patterns of mean velocity and turbulent airflow are reported for isolated barchan dunes. Turbulence was sampled using a high frequency sonic anemometer, deriving near‐surface Reynolds shear and normal stresses. Measurements upwind of and over a crest‐brink separated barchan indicated that shear stress was sustained despite a velocity reduction at the dune toe. The mapped streamline angles and enhanced turbulent intensities suggest the effects of positive streamline curvature are responsible for this maintenance of shear stress. This field evidence supports an existing model for dune morphodynamics based on wind tunnel turbulence measurements. Downwind, the effect of different dune profiles on flow re‐attachment and recovery was apparent. With transverse incident flow, a re‐attachment length between 2·3 and 5·0h (h is dune brink height) existed for a crest‐brink separated dune and 6·5 to 8·6h for a crest‐brink coincident dune. The lee side shear layer produced elevated turbulent stresses immediately downwind of both dunes, and a decrease in turbulence with distance characterized flow recovery. Recovery of mean velocity for the crest‐brink separated dune occurred over a distance 6·5h shorter than the crest‐brink coincident form. As the application of sonic anemometers in aeolian geomorphology is relatively new, there is debate concerning the suitability of processing their data in relation to dune surface and streamline angle. This paper demonstrates the effect on Reynolds stresses of mathematically correcting data to the local streamline over varying dune slope. Where the streamline angle was closely related to the surface (windward slope), time‐averaged shear stress agreed best with previous wind tunnel findings when data were rotated along streamlines. In the close lee, however, the angle of downwardly projected (separated) flow was not aligned with the flat ground surface. Here, shear stress appeared to be underestimated by streamline correction, and corrected shear stress values were less than half of those uncorrected. Copyright © 2011 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.     

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.     

Conditional statistics of Reynolds shear stress in flow over 2D dunes in the presence of surface waves

This study presents the analysis of the velocity fluctuations to describe the conditional statistics of Reynolds shear stress in flow over two‐dimensional dunes in the presence of surface waves of varying frequency. The flow velocity measurements over the dunes are made using a 16‐MHz 3D acoustic Doppler velocimeter. The joint probability distributions of the normalized stream‐wise and vertical velocity fluctuations at different vertical locations are calculated in the trough region of a selected dune in quasi‐steady region of the flow. Third‐order moments of the stream‐wise and vertical velocity components over one dune length are also calculated throughout the flow depth for understanding the effect of surface waves on relative contributions to the Reynolds shear stress due to the four quadrant events. The structure of instantaneous Reynolds stresses is analysed using quadrant analysis technique. It has been shown that the contributions of second and fourth quadrant events to the Reynolds shear stress increase with increase in the frequency of surface waves. In fact, the largest contribution to turbulent stresses comes from the second quadrant. The cumulant discard method is applied to describe the statistical properties of the covariance term u′w′. Conditional statistics and conditional sampling are used to compare the experimental and theoretical relative contributions to the Reynolds shear stress from the four quadrant events. Copyright © 2013 John Wiley & Sons, Ltd.     

Influence of bank vegetation and gravel bed on velocity and Reynolds stress distributions

This paper presents the results of a laboratory flume experimental study on the interaction of bank vegetation and gravel bed on the flow velocity （primarily on the location of the maximum velocity, Umax） and the Reynolds stress distributions. The results reveal that the dip of the maximum velocity below the water surface is up to 35% of flow depth and the difference between Umax and the velocity at the water surface is considerable in the presence of vegetation on the walls. The zone of the log-law varies from y/h=2 up to 15 percent of flow depth and it does not depend on distance from the wall. Deviation of the velocity profile in the outer layer over a gravel bed with vegetation cover on the walls is much larger than the case of flow over a gravel bed without vegetation cover on the walls. The presence of vegetation on the walls changes uniform flow to non-uniform flow. This fact can be explained by considering the nonlinear Reynolds stress distribution and location of maximum velocity in each profile at different distances across the flume. The Reynolds stress distributions at the distance 0.02 m from the wall have negative values and away from the wall, they change the sign taking positive values with specific convex form with apex in higher location. Average of von Karman constant κ for this study is equal to 0.16. Based on to=0.16, the methods of Clauser and the Reynolds stress are compatible for determination of shear velocity.     

Laboratory flume studies of microflow environments of aquatic plants

The microflow environments of aquatic plants with reference to Myriophyllum and Hydrilla are simulated in a laboratory flume. A Nix Streamflow microflow meter was used to measure the mean velocity profiles of flow at different densities of plants, flow ranges and measurement positions. Each mean velocity profile consists of three hydrodynamic regimes (i.e. within‐canopy zone, above‐canopy zone and a transitional zone between them), which indicate the presence of two benthic boundary layers (internal and external ones). Out of 38 measured mean velocity profiles, most do not fit a logarithmic relationship. The following hydrodynamic parameters are used in characterizing the flow regimes: local shear velocity (u_*), roughness length (z_o), canopy roughness Reynolds number (Re_*), bed shear stress (τ_o) and laminar sublayer (σ). Copyright © 2002 John Wiley & Sons, Ltd.     

Examining the physical components of boundary shear stress for water‐worked gravel deposits

It is argued in this commentary that, in order to understand better the physical mechanisms that generate boundary shear stress over water‐worked gravel beds, flow velocity data should be re‐evaluated by spatial averaging the Reynolds equations to produce time‐ and space‐averaged (double‐averaged) momentum equations. A series of laboratory experiments were conducted in which the flow velocities were measured using a PIV system over two water‐worked gravel deposits. Combined with detailed data on the bed surface topography and vertical porosity, the physical components of shear stress were obtained. This enabled the various momentum transfer mechanisms present above, within and at the interface of a porous, fluvial deposit, to be quantified. This included the examination of the relevant contributions of temporal and spatial fluctuations in velocity and surface drag to the overall momentum transfer. It is demonstrated that double‐averaging represents a logical framework for assessing the fluid forces responsible for sediment entrainment and for investigating intragravel flow and sediment–water interface exchange mechanisms within the roughness layer in water‐worked gravel deposits. By considering the physical components of shear stress and their relative sizes it was possible to provide a physically based explanation for existing observations of enhanced mobility of gravel–sand mixtures and the transfer of solutes into porous, gravel deposits. This analysis reveals the importance of obtaining co‐located, high quality spatial data on the flow field and bed surface topography in order to gain a physical understanding of the mechanisms which generate boundary shear stress. Copyright © 2010 John Wiley & Sons, Ltd.     

Dimensionless critical shear stress evaluation from flume experiments using different gravel beds

Flume experiments were conducted using four different gravel beds (D₅₀ + 12–39 mm) and a range of marked particles (10–65 mm). The shear stresses were evaluated from friction velocities, when initial movement of marked particles occurred. Two kinds of equations were produced: first for the threshold of initial movement, and second for generalized movement. Equations of the type 0_c + a(D_i/D₅₀)^b, as proposed by Andrews (1983) are applicable even if the material is relatively well sorted. However, the values of a and b are lower (respectively 0·050 and -0·70) for initial movement. Generalized movement requires a higher shear stress (a + 0·068 and b + -0·80). D₉₀ of the bed material and y₀ (the bed roughness parameter) were also used as reference values in place of D₅₀. They produced lower values than in natural streams, mainly owing to the fact that the material used in the flume is better sorted: clusters are less well developed and the bed roughness is lower.     

Velocity distribution in a gradually accelerating free surface flow

This study investigates the interaction of the vertical velocity v and the streamwise velocity u in a gradually accelerating flow. The analytical result shows that the momentum of uv driven by the mean velocities in a non-uniform flow is not negligible. This additional momentum directly results in the concave profiles of Reynolds shear stress in gradually accelerating flows, a departure from the expected linear profile. Consequently, this momentum causes the maximum velocity to be located below the free surface, i.e., the dip-phenomenon. This paper investigated the interactions of the Reynolds shear stress, non-zero vertical velocity and dip-phenomenon, it is found that the non-zero vertical velocity causes the dip-phenomenon. The approach is tested using the experimental data of Song and others, and good agreements between the predicted and measured velocity profiles have been achieved.     

Relation between flow,surface‐layer armoring and sediment transport in gravel‐bed rivers

This study investigates trends in bed surface and substrate grain sizes in relation to reach‐scale hydraulics using data from more than 100 gravel‐bed stream reaches in Colorado and Utah. Collocated measurements of surface and substrate sediment, bankfull channel geometry and channel slope are used to examine relations between reach‐average shear stress and bed sediment grain size. Slopes at the study sites range from 0·0003 to 0·07; bankfull depths range from 0·2 to 5 m and bankfull widths range from 2 to 200 m. The data show that there is much less variation in the median grain size of the substrate, D_50s, than there is in the median grain size of the surface, D₅₀; the ratio of D₅₀ to D_50s thus decreases from about four in headwater reaches with high shear stress to less than two in downstream reaches with low shear stress. Similar trends are observed in an independent data set obtained from measurements in gravel‐bed streams in Idaho. A conceptual quantitative model is developed on the basis of these observations to track differences in bed load transport through an idealized stream system. The results of the transport model suggest that downstream trends in total bed load flux may vary appreciably, depending on the assumed relation between surface and substrate grain sizes. Copyright © 2007 John Wiley & Sons, Ltd.     

Flow modelling in gravel‐bed rivers: rethinking the bottom boundary condition

A key problem in computational fluid dynamics (CFD) modelling of gravel‐bed rivers is the representation of multi‐scale roughness, which spans the range from grain size, through bedforms, to channel topography. These different elements of roughness do not clearly map onto a model mesh and use of simple grain‐scale roughness parameters may create numerical problems. This paper presents CFD simulations for three cases: a plane bed of fine gravel, a plane bed of fine gravel including large, widely‐spaced pebble clusters, and a plane gravel bed with smaller, more frequent, protruding elements. The plane bed of fine gravel is modelled using the conventional wall function approach. The plane bed of fine gravel including large, widely‐spaced pebble clusters is modelled using the wall function coupled with an explicit high‐resolution topographic representation of the pebble clusters. In these cases, the three‐dimensional Reynolds‐averaged continuity and Navier–Stokes equations are solved using the standard k ? ε turbulence model, and model performance is assessed by comparing predicted results with experimental data. For gravel‐bed rivers in the field, it is generally impractical to map the bed topography in sufficient detail to enable the use of an explicit high‐resolution topography. Accordingly, an alternative model based on double‐averaging is developed. Here, the flow calculations are performed by solving the three‐dimensional double‐averaged continuity and Navier‐Stokes equations with the spatially‐averaged 〈k ? ε〉 turbulence model. For the plane bed of fine gravel including large, widely‐spaced pebble clusters, the model performance is assessed by comparing the spatially‐averaged velocity with the experimental data. The case of a plane gravel bed with smaller, more frequent, protruding elements is represented by a series of idealized hypothetical cases. Here, the spatially‐averaged velocity and eddy viscosity are used to investigate the applicability of the model, compared with using the explicit high‐resolution topography. The results show the ability of the model to capture the spatially‐averaged flow field and, thus, illustrate its potential for representing flow processes in natural gravel‐bed rivers. Finally, practical data requirements for implementing such a model for a field example are given. Copyright © 2011 John Wiley & Sons, Ltd.