Saturated,collisional flows of spheres over an inclined,erodible bed between vertical sidewalls

We consider a steady, uniform, dense, fully-saturated, gravity-driven, inclined flow of water and identical spherical grains over an erodible bed between parallel, vertical sidewalls. The grains are inelastic and the energy lost in their interaction is also influenced by the fluid viscosity. We use an extension of kinetic theory for dense flows and employ approximate integrations of the momentum and energy balances for the grains in order to obtain analytical expressions for the depth of flow and for the volume flow rate of the mixture as functions of inclination and average concentration. We also predict the range of slopes for which dense, fully-saturated flows are possible. The predictions are in reasonable agreement with already published experimental results.     

Reinvestigation on mixing length in an open channel turbulent flow

The present study proposes a model on vertical distribution of streamwise velocity in an open channel turbulent flow through a newly proposed mixing length, which is derived for both clear water and sediment-laden turbulent flows. The analysis is based on a theoretical consideration which explores the effect of density stratification on the streamwise velocity profile. The derivation of mixing length makes use of the diffusion equation where both the sediment diffusivity and momentum diffusivity are taken as a function of height from the channel bed. The damping factor present in the mixing length of sediment-fluid mixture contains velocity and concentration gradients. This factor is capable of describing the dip-phenomenon of velocity distribution. From the existing experimental data of velocity, the mixing length data are calculated. The pattern shows that mixing length increases from bed to the dip-position, having a larger value at dip-position and then decreases up to the water surface with a zero value thereat. The present model agrees well with these data sets and this behavior cannot be described by any other existing model. Finally, the proposed mixing length model is applied to find the velocity distribution in wide and narrow open channels. The derived velocity distribution is compared with laboratory channel data of velocity, and the comparison shows good agreement.     

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.     

Numerical calculation for bed variation in compound-meandering channel using depth integrated model without assumption of shallow water flow

In a compound meandering channel, patterns of flow structures and bed variations change with increasing water depth owing to complex momentum exchange between high-velocity flow in a main channel and low-velocity flows in flood plains. We have developed a new quasi-three-dimensional model without the shallow water assumption, i.e., hydrostatic pressure distribution; our method is known as the general bottom velocity computation (BVC) method. In this method, a set of depth-integrated equations, including depth-integrated momentum and vorticity equations, are prepared for evaluating bottom velocity and vertical velocity distributions. The objective of this study is to develop a bed variation calculation method for both single and compound meandering channels by using the BVC method coupled with a sediment transport model. This paper shows that the BVC method can reproduce the pattern change of bed variation in a compound meandering channel flow with increasing relative depth. The variation in sediment transport rate due to overbank flow is explained by experimental and computational results.     

EXPERIMENTAL STUDY ON EQUILIBRIUM CONCENTRATION OF DEBRIS FLOWS

Discussion open until 2002.EmunmL SwrY ONEQUII.thare CONCENTRATION OF DEBRIS rrOWSBin YU1AaSTsiCTThe paPe PresentS experimntal study of debris flows. The equiMm concentIation of solidparticle in the now is a hahon of the energy slope, density of solid Particle and kinetic ffichonange of paxtiles. The kinhc forhon angle is a funhon of intemal ffichon angle, the cOnCentrationof solid phocles and the mtalmum POssible concewhon. TO deteImin th6 hahon between thekinetic fficho…     

Spatial variability of three‐dimensional Reynolds stresses in a developing channel bend

Experimental results of the mean flow field and turbulence characteristics for flow in a model channel bend with a mobile sand bed are presented. Acoustic Doppler velocimeters (ADVs) were used to measure the three components of instantaneous velocities at multiple cross sections in a 135° channel bend for two separate experiments at different stages of clear water scour conditions. With measurements at multiple cross sections through the bend it was possible to map the changes in both the spatial distribution of the mean velocity field and the three Reynolds shear stresses. Turbulent stresses are known to contribute to sediment transport and the three‐dimensionality inherent to flow in open channel bends presents a useful case for determining specific relations between three‐dimensional turbulence and sediment entrainment and transport. These measurements will also provide the necessary data for validating numerical simulations of turbulent flow and sediment transport. The results show that the magnitude and distribution of three‐dimensional Reynolds stresses increase through the bend, with streamwise‐cross stream and cross stream‐vertical components exceeding the maximum principal Reynolds stress through the bend. The most intriguing observation is that near‐bed maximum positive streamwise‐cross stream Reynolds stress coincides with the leading edge of the outer bank scour hole (or thalweg), while maximum cross stream‐vertical Reynolds stress (in combination with high negative streamwise‐cross stream Reynolds stress near the bend apex) coincides with the leading edge of the inner bank bar. Maximum Reynolds stress and average turbulent kinetic energy appear to be greater and more localized over the scour hole before final equilibrium scour is reached. This suggests that the turbulent energy in the flow is higher while the channel bed is developing, and both lower turbulent energy and a broader distribution of turbulent stresses near the bed are required for cessation of particle mobilization and transport. Copyright © 2010 John Wiley & Sons, Ltd.     

Experimental investigation of the velocity of a sand cloud blowing over a sandy surface

The velocity of a wind‐blown sand cloud is important for studying its kinetic energy, related erosion, and control measures. PDA (particle dynamics analyser) measurement technology is used in a wind tunnel to study the probability distribution of particle velocity, variations with height of the mean velocity and particle turbulence in a sand cloud blowing over a sandy surface. The results suggest that the probability distribution of the particle velocity in a blowing sand cloud is stochastic. The probability distribution of the downwind velocity complies with a Gaussian function, while that of the vertical velocity is greatly complicated by grain impact with the bed and particle–particle collisions in the air. The probability distribution of the vertical velocity of ?ne particles (0·1–0·3 mm sands) can be expressed as a Lorentzian function while that of coarse particles (0·3–0·6 mm sands) cannot be expressed by a simple distribution function. The mean downwind velocity is generally one or two orders greater than the mean vertical velocity, but the particle turbulence in the vertical direction is at least two orders greater than that in the downwind direction. In general, the mean downwind velocity increases with height and free‐stream wind velocity, but decreases with grain size. The variation with height of the mean downwind velocity can be expressed by a power function. The particle turbulence of a blowing sand cloud in the downwind direction decreases with height. The variations with height of the mean velocity and particle turbulence in the vertical direction are very complex. It can be concluded that the velocity of a sand cloud blowing over a sandy surface is mainly in?uenced by wind velocity, grain impact with the bed and particle–particle collisions in the air. Wind velocity is the primary factor in?uencing the downwind velocity of a blowing sand cloud, while the grain impact with the bed and particle–particle collisions in the air are the primary factors responsible for the vertical velocity. Copyright © 2004 John Wiley & Sons, Ltd.     

A worldwide correlation for exponential bed particle size variation in subaerial aqueous flows

The particle size of the bed sediments in or on many natural streams, alluvial fans, laboratory flumes, irrigation canals and mine waste deltas varies exponentially with distance along the stream. A plot of the available worldwide exponential bed particle size diminution coefficient data against stream length is presented which shows that all the data lie within a single narrow band extending over virtually the whole range of stream lengths and bed sediment particle sizes found on Earth. This correlation applies to both natural and artificial flows with both sand and gravel beds, irrespective of either the solids concentration or whether normal or reverse sorting occurs. This strongly suggests that there are common mechanisms underlying the exponential diminution of bed particles in subaerial aqueous flows of all kinds. Thus existing models of sorting and abrasion applicable to some such flows may be applicable to others. A comparison of exponential laboratory abrasion and field diminution coefficients suggests that abrasion is unlikely to be significant in gravel and sand bed streams shorter than about 10 km to 100 km, and about 500 km, respectively. Copyright © 1999 John Wiley & Sons, Ltd.     

Submerged granular channel flows driven by gravity

The paper presents a rheological model for gravity driven granular flows saturated with water. The model adopts the kinetic theory for the collisional regime, which is dominant near the free surface, while for the frictional regime a specific model is proposed, which matches the Coulombian condition at the boundary with the loose static bed. The solution for the frictional regime is based on the observation that the frictional and the collisional regimes are not stratified but coexist across the flow depth.The model is able to predict the distribution along the depth of velocity, concentration, granular temperature, shear and normal stresses. In particular, it is possible to discriminate between the collisional and the frictional components of the normal and shear stresses.The results of the model are compared with the data of a laboratory investigation on a steady, uniform, highly concentrated saturated granular flow, composed of spheres with a uniform diameter of 6 mm.Another important issue addressed in the paper concerns the balances of the kinetic energy of the granular phase. The model is able to describe the mechanisms of production, diffusion and dissipation of kinetic energy, relevant to both the mean component of the flow and the fluctuating component (i.e., the collisional component). Also in this case the comparison with the experimental data is reasonably good. Near the static loose bed, the model predicts that the flux of the diffused fluctuating energy exceeds an order of magnitude the locally dissipated flux of fluctuating energy. This suggests that the motion of the grains, even at concentrations close to that of packing, is always accompanied by a certain degree of granular temperature.     

STOCHASTIC THEORY OF THE DISTRIBUTION OF SEDIMENT PARTICLES SUSPENDED IN A TURBULENT FLOW

The random motion of sediment particles suspended in a turbulent flow is studied by means of stochastic process. Results of analysis of particle's frequency response to the random force exerted on the particle due to fluid turbulence suggest that only the lower part of the whole frequency range of the eddy motion will govern the motion of the particle. The mean values of particle velocity and displacement in the vertical direc- tion are calculated and it is found that particle velocity vp- can be decomposed into a mean motion and a velocity fluctuation vp- , where is equal to the fall velocity in tranquil fluid. An Ito^ random differential equation for particle dis- placement Yp is developed, from which a Fokker-Planck equation for the probability density function p(y,t) is derived on the basis of the theory of Markov process, where y denotes the vertical coordinate. The vertical distribution of the particle is thus interrelated to the random motion of the particle. The an effect that a particle will be subject to in the neighborhood or the bottom boundary is taken into consideration and a corresponding Fokker-Planck equation is developed. Analytical solution of the Fok- ker-Planck equation including the lift force effect shows that probability density p(y,t) for the particle displacement has a maximum value at y = H where the perpen- dicular component of the lift force balances the particle gravity. This theoretical result agrees with experimental observations as reported in literature.     

Velocity and concentration profiles of particle movement in sheet flows

Sheet flows occur widely in natural free-surface flows including rivers in flood, tidal estuaries and coastal waters in storm conditions when bed shear stress becomes sufficiently high. Particle volumetric concentration in sheet flows normally follows a linear distribution with the Rouse [Rouse H. Modern conceptions of the mechanics of fluid turbulence. Trans ASCE, 1937;102:463–543] distribution applicable in the dilute water column above the sheet-flow layer. However, a well-verified formula for determining particle velocity distribution in sheet flows is still lacking. This paper presents formulas to describe the particle velocity profile in steady or oscillatory sheet flows. They compare well with measured data. In particular, the novel formula for determining the particle velocity at the top of bedload–sediment-dominated sublayer in sheet flows is also well verified with measured data.     

A Bed-load function based on kinetic theory

Bed-load transport plays a fundamental role in morphological processes of natural rivers and human-made channels.This paper presents bed-load function derived on the basis of kinetic theory. The bed-load function is obtained by integrating the pick-up rate of bed sediment with respect to its longitudinal travel distance,following the basic concept on bed-load put forward by Einstein.The pick-up rate is expressed as an upwards flux of bed sediment and determined by invoking the particle velocity distribution function derived by solving the Boltzmann equation of kinetic theory. Comparisons of the present formula with six other bed-load formulas and the experimental data are also made in this paper.The results show that the present bed-load formula agrees well with the experimental data.In addition,the influences due to collision between particles on bed-load is discussed which shows that an appreciable damping of the intensities of bed-load is observed only when the shear acting on particles is large enough to increase the concentration and intensity of random motion of bed load particles so that the collisions between sediment particles can occur.     

Turbulent mechanisms in open channel sediment-laden flows

The effects of turbulence on water-sediment mixtures is a critical issue in studying sediment-laden flows. The sediment concentrations and particle inertia play a significant role in the effects of turbulence on mixtures. A two-phase mixture turbulence model was applied to investigate the turbulence mechanisms affecting sediment-laden flows. The two-phase mixture turbulence model takes into account the complicated mechanisms arising from interphase transfer of turbulent kinetic energy, particle collisions, and stratification. The turbulence in sediment-laden flows is the result of the interaction of four factors, i.e. the production, dissipation, diffusion, and inter-phase transfer of turbulent kinetic energy of mixtures. The turbulence production and dissipation are two dominant processes which balance the turbulent kinetic energy of mixtures. The turbulence production represents turbulence intensity, while the inter-phase transfer of turbulent kinetic energy denotes the effect of particles on the turbulence of sediment-laden flows. Although, the magnitude of the inter-phase interaction term is much less than that of the turbulence production and dissipation terms, due to an approximate local balance between production and dissipation of the turbulent kinetic energy, even the small order of the inter-phase interaction has a significant impact on the turbulent balance of sediment-laden flows. The presence of particles plays a duel role in the turbulence dissipation of mixtures: both promotion and suppression. An important parameter used to determine the turbulent viscosity of mixtures, which is constant in clear water, is the function of the sediment concentration and particle inertia in sediment-laden flows.     

ON THE VERTICAL DISTRIBUTION OF SUSPENDED SEDIMENT CONCENTRATION

The problem of sediment concentration distribution in a vertical for channel and pipe flows is studied in this paper. Considering the variation of the vertical component of the fluctuation velocity v as an independent variable, two types of sediment concentration distribution can be derived. However, only one type of distribution is commonly quoted while the other one rarely reported. In this paper explanation for such phenomena is given and conditions for the occurrence of both types are also discussed.     

A depth-averaged two-phase model for debris flows over fixed beds

A depth-averaged two-phase model is proposed for debris flows over fixed beds, explicitly incorporating interphase and particle-particle interactions, fluid and solid fluctuations and multi grain sizes. A first-order model based on the kinetic theory of granular flows is employed to determine the stresses due to solid fluctuations, while the turbulent kinetic energy - dissipation rate model is used to determine the stresses from fluid fluctuations. A well-balanced numerical algorithm is applied to solve the governing equations. The present model is benchmarked against USGS experimental debris flows over fixed beds. Incorporating the stresses due to fluid and solid fluctuations and properly estimating the bed shear stresses are shown to be crucial for reproducing the debris flows. Longitudinal particle segregation is resolved, demonstrating coarser sediments around the fronts and finer grains trailing the head. Based on extended modeling exercises, debris flow efficiency is shown to increase with initial volume, which is underpinned by observed datasets.     

Bed scour by debris flows: experimental investigation of effects of debris‐flow composition

Debris flows can grow greatly in size by entrainment of bed material, enhancing their runout and hazardous impact. Here, we experimentally investigate the effects of debris‐flow composition on the amount and spatial patterns of bed scour and erosion downstream of a fixed to erodible bed transition. The experimental debris flows were observed to entrain bed particles both grain by grain and en masse, and the majority of entrainment was observed to occur during passage of the flow front. The spatial bed scour patterns are highly variable, but large‐scale patterns are largely similar over 22.5–35° channel slopes for debris flows of similar composition. Scour depth is generally largest slightly downstream of the fixed to erodible bed transition, except for clay‐rich debris flows, which cause a relatively uniform scour pattern. The spatial variability in the scour depth decreases with increasing water, gravel (= grain size) and clay fraction. Basal scour depth increases with channel slope, flow velocity, flow depth, discharge and shear stress in our experiments, whereas there is no correlation with grain collisional stress. The strongest correlation is between basal scour and shear stress and discharge. There are substantial differences in the scour caused by different types of debris flows. In general, mean and maximum scour depths become larger with increasing water fraction and grain size, and decrease with increasing clay content. However, the erodibility of coarse‐grained experimental debris flows (gravel fraction = 0.64) is similar on a wide range of channel slopes, flow depths, flow velocities, discharges and shear stresses. This probably relates to the relatively large influence of grain‐collisional stress to the total bed stress in these flows (30–50%). The relative effect of grain‐collisional stress is low in the other experimental debris flows (<5%), causing erosion to be largely controlled by basal shear stress. Copyright © 2016 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.     

Particle size distribution of bed materials in the sandy river bed of alluvial rivers

The particle size distribution of bed materials in the sandy river bed of alluvial rivers is important in the study of topics such as friction, river bed evolution, erosion, and siltation. It also can reflect the dependency relation between river bed sediment and flow intensity. In this paper, the critical pattern of sediment movement in the near-wall region of a sandy river bed was analyzed. According to the principle of momentum balance, the critical settling-rising condition of bed material in a sandy river bed was found to be instantaneous turbulent velocity equal to 2.7 times the sediment settling velocity in quiescent water. Based on a vertical instantaneous turbulent velocity with a Gaussian distribution, a theoretical relation for calculating the particle size distribution of bed materials in a sandy river bed without pre-known characteristic grain sizes was developed by solving a stochastic equation. The for-mula is verified using measured data, and the results show that the proposed formula was in accordance with the measured data. This study has theoretical significance and practical value for determining the bed material particle size distribution of the sandy bed of alluvial rivers.     

Time-mean structure of secondary flows in open channel with longitudinal bedforms

Secondary motions are commonly present in open channel flows. This study aims to investigate, both experimentally and analytically, time-mean characteristics of cellular secondary flows generated by longitudinal bedforms. Experiments were conducted in a tilting, rectangular flume with six different longitudinal bedforms, including alternate bed strips with different roughness heights and bed ridges of wavy and rectangular shapes. Various flows were sampled using a two-dimensional laser Doppler anemometer (LDA) and a one-dimensional ultrasonic Doppler velocimeter (UDV). Experimental results demonstrate secondary flows appearing basically in cellular fashion over the modeled longitudinal bedforms. It is also shown that the cellular structures can be described analytically with kinematic considerations. The discrepancies between theoretical and measurement results are discussed. An empirical relationship between maximum vertical velocity and bed configuration is finally proposed based on the experimental data.     

Significance of particle interaction to the modelling of cohesive sediment transport in rivers

Aggregation processes of fine sediments have rarely been integrated in numerical simulations of cohesive sediment transport in riverine systems. These processes, however, can significantly alter the hydrodynamic characteristics of suspended particulate matter (SPM), modifying the particle settling velocity, which is one of the most important parameters in modelling suspended sediment dynamics. The present paper presents data from field measurements and an approach to integrate particle aggregation in a hydrodynamic sediment transport model. The aggregation term used represents the interaction of multiple sediment classes (fractions) with corresponding multiple deposition behaviour. The k–ε–turbulence model was used to calculate the coefficient of vertical turbulent mixing needed for the two‐dimensional vertical‐plane simulations. The model has been applied to transport and deposition of tracer particles and natural SPM in a lake‐outlet lowland river (Spree River, Germany). The results of simulations were evaluated by comparison with field data obtained for two levels of river discharge. Experimental data for both discharge levels showed that under the prevailing uniform hydraulic conditions along the river reach, the settling velocity distribution did not change significantly downstream, whereas the amount of SPM declined. It was also shown that higher flow velocities (higher fluid shear) resulted in higher proportions of fast settling SPM fractions. We conclude that in accordance with the respective prevailing turbulence structures, typical aggregation mechanisms occur that continuously generate similar distribution patterns, including particles that settle toward the river bed and thus mainly contribute to the observed decline in the total SPM concentration. In order to determine time‐scales of aggregation and related mass fluxes between the settling velocity fractions, results of model simulations were fitted to experimental data for total SPM concentration and of settling velocity frequency distributions. The comparison with simulations for the case of non‐interacting fractions clearly demonstrated the practical significance of particle interaction for a more realistic modelling of cohesive sediment and contaminant transport. Copyright © 2004 John Wiley & Sons, Ltd.