Equilibrium Bed Load Transport Research Articles

Theoretical derivation of hydraulic geometry equations for a gravel bed river channel

The geomorphic relations of hydraulic geometry at-a-station and downstream, expressed as power laws, are derived for a straight, single thread, gravel river channel. The cross-section has a plane central bed, narrow or wide, flanked by curved sides or banks. The analysis employs mechanistic formulae for fixed and mobile bed flow resistance, downchannel bedslope, bedload transport and the novel introduction of bedload ratings all expressed as power laws. Steady uniform flow and equilibrium bedload transport conditions are assumed to occur above the central section. Water discharge and sediment size are assumed to be the only known variables. Exponents of the relations of hydraulic geometry are fixed and are consistent with reported theoretical estimates and field measurements. Coefficients are also explicitly derived and are consistent with field measurements and vary owing to the presence of independent variables in their definition. Here aspect ratio has the major influence. The derived equations can provide coarse estimates for the design of canal and river control works. Conditional separate relations between both width and depth, and bed material size or bed slope are also determined for the occurrence of at-many-stations hydraulic geometry. To improve generality, future work should consider the hydraulic geometry of straight gravel channels with bedload transport, but having an irregular cross-section.

Effect of Increasing Antecedent Flows on Equilibrium Bed-Load Transport Rates in a Laboratory Channel with a Sand and Gravel Bed Channel

Experiments were conducted in a laboratory flume in which gravel and total bed-load rates were measured continuously using independent methods that allowed evaluation of the effect of four antecedent flow conditions on the transport of bed load under standard flow. It was found that both mean total bed-load rate and gravel transport rates are related to the magnitude of antecedent flow. The changes in mean bed-load rates were attributed to changes in the stability of the coarsest fractions of the bed material, caused by the previous high-flow event. This work indicates that long-term mean bed-load transport rates for a sand and gravel bed are not just a function of grain size and flow rate but also vary with the magnitude of antecedent flow.

A 3-D finite volume model for sediment transport in coastal waters

A 3-D model has been developed to simulate sediment transport and bed change induced by currents and waves in coastal waters. The currents are calculated with the 3-D phase-averaged shallow water flow equations, the wave characteristics are determined with a horizontal 2-D wave spectral transformation model, and multiple-sized non-cohesive suspended load and bed load are simulated using a non-equilibrium transport model. The classical mixing length model is modified to determine the horizontal and vertical eddy viscosity considering the effects of current, waves, and wave breaking. A new approach is proposed in the present 3-D model framework to calculate the bed grain shear stress, which is applied to determine the equilibrium bed-load transport rate and near-bed suspended-load concentration. The flow and sediment transport equations are numerically solved using an implicit finite volume method on a hexahedral mesh constructed with horizontal telescoping (quadtree) rectangular cells and vertical sigma coordinate. The difference in the near-bed cells for flow and suspended load calculations is avoided by adding the bed load to the suspended load in the first flow cell above the bed. The developed model has been tested in four laboratory and field cases. It reasonably well reproduces the measured water levels, flow velocities, sediment concentrations, and bed changes.

Flow resistance law under equilibrium bed-load transport conditions

The uniform flow resistance equation, in the form due to Manning or Darcy-Weisbach, is often applied to determine the stage-discharge relationship of a river cross-section. The application of this equation, namely the slope-area method, allows to indirectly measure by water level readings the corresponding river discharge. In this paper, a recently deduced flow resistance equation for open channel flow was tested during conditions of equilibrium bed-load transport. First the flow resistance equation was determined by dimensional analysis and applying the condition of incomplete self-similarity for the flow velocity profile. Then the analysis was developed by the following steps: (i) the flow resistance equation was calibrated by available laboratory measurements of flow velocity, water depth and bed slope carried out in 143 flume experimental runs during conditions of equilibrium bed-load transport; (ii) a relationship (Eq. (33)) between the Γ function of the velocity profile, the channel slope, the Shields number and the Froude number was established by the available measurements; (iii) the relationship for estimating the Γ function (Eq. (33)) and the theoretical flow resistance equation were tested by 127 independent flume measurements carried out for flows with bedload transport (iv) Eq. (33) and the flow resistance law were tested by field measurements carried out in 104 reaches of some Calabrian gravel bed rivers; and (v), finally, the relationship for estimating the Γ function was recalibrated (Eq. (34)) using all laboratory measurements (270 experimental runs).The proposed theoretical resistance law, calibrated by the measurements of Recking et al. (2008), allowed an estimate of the Darcy-Weisbach friction factor which is more accurate than that obtained by the original approach of Recking et al.The theoretical flow resistance law (Eq. (28)) coupled with the relationship for estimating the Γ function (Eq. (34)), which is characterized by the applicability of a wide range of flow conditions, allowed to estimate the Darcy-Weisbach friction factor for flows with and without bed-load.

Quantification of spatial lag effect on sediment transport around a hydraulic structure using Eulerian–Lagrangian model

The spatial lag effect on sediment transport around a hydraulic structure is quantified through numerical simulation using a Eulerian–Lagrangian model. The Eulerian–Lagrangian model uses a deterministic approach of bed load motion and stochastic approach of Einstein's concept, which allows for a reasonable quantification of both non-equilibrium and equilibrium bed load transport rates. Numerical simulations are conducted for local scour around a spur dyke and a weir-type structure. As pointed out in previous experimental studies on the one-dimensional problem, Eulerian–Lagrangian simulation reveals an unsaturated bed load in a scour hole around a hydraulic structure. Furthermore, an oversaturated bed load is also found in the deposition region, particularly behind the structure. The validity of the linear adaptation equation is discussed using numerical results, revealing the possibility that the linear adaptation equation does not appropriately represent the existence of the strong spatial lag effect. The present study finds there are two modes in the response of the adaptation length to the unsaturated bed load, which depend on the clear-water scour condition or the live-bed scour condition. Findings of this study will contribute to multi-dimensional sediment transport simulations that consider the spatial lag effect.

Modeling sediment transport with an integrated view of the biofilm effects

AbstractMost natural sediment is invariably covered by biofilms in reservoirs and lakes, which have significant influence on bed form dynamics and sediment transport, and also play a crucial role in natural river evolution, pollutant transport, and habitat changes. However, most models for sediment transport are based on experiments using clean sediments without biological materials. In this study, a three‐dimensional mathematical model of hydrodynamics and sediment transport is presented with a comprehensive consideration of the biofilm effects. The changes of the bed resistance mainly due to the different bed form dynamics of the biofilm‐coated sediment (biosediment), which affect the hydrodynamic characteristics, are considered. Moreover, the variations of parameters related to sediment transport after the biofilm growth are integrated, including the significant changes of the incipient velocity, settling velocity, reference concentration, and equilibrium bed load transport rate. The proposed model is applied to evaluate the effects of biofilms on the hydrodynamic characteristics and sediment transport in laboratory experiments. Results indicate that the mean velocity increases after the biofilm growth, and the turbulence intensity near the river bed decreases under the same flow condition. Meanwhile, biofilm inhibits sediment from moving independently. Thus, the moderate erosion is observed for biosediment resulting in smaller suspended sediment concentrations. The proposed model can reasonably reflect these sediment transport characteristics with biofilms, and the approach to integration of the biological impact could also be used in other modeling of sediment transport, which can be further applied to provide references for the integrated management of natural aqueous systems.

Flume Experiments to Constrain Bedload Adaptation Length

Spatially variable channel geometry in natural rivers produces nonuniform flow and spatial gradients in the shear stress field. The travel distance required for the flow to acquire the capacity bedload concentration and attain a new equilibrium bedload transport rate upon encountering a region of higher or lower shear stress is defined as the bedload adaptation length (Lb). Estimates of Lb are used by some numerical morphodynamic models to account for nonequilibrium bedload transport in the computation of local transport rates. However, current methods for estimating this parameter are uncertain and often crude. The authors therefore conducted experiments designed to measure Lb for a uniform sediment mixture in a laboratory flume. Instantaneous bedload transport rates were determined by counting passing sediment particles on digital imagery collected at variable distances downstream from a zero-transport boundary in a small flume. The flume was operated at three bed slopes in order to assess Lb over a range of hydraulic conditions. Bedload adaptation length was found to be about 30±8 particle diameters at a relatively low excess dimensionless shear stress (θ−θc=0.018, where θ is the dimensionless shear stress and θc=0.0436 is the critical dimensionless shear stress) and about 100±30 particle diameters at a moderate level of excess dimensionless shear stress (θ−θc=0.032). The experiments failed to resolve Lb at higher shear stresses. These results support physically based models that cast Lb as an increasing function of excess shear stress. They also suggest that Lb may be small relative to the resolution of the numerical mesh used in many modeling applications. In such cases, model performance may be insensitive to the choice of any arbitrary small value of Lb.

Experiments on dispersion of tracer stones under lower‐regime plane‐bed equilibrium bed load transport

A common approach for estimating the bed load transport rate in gravel‐bed streams is to relate it to deterministic channel‐averaged driving parameters and corresponding resistance properties of the bed material. Notwithstanding the proven success of this approach in modeling various morphodynamic scenarios, it does not contain the mechanics necessary to relate the bulk sediment transport rate to the displacement patterns of individual particles. Experiments on entrainment, transport, and deposition of tracer stones in a flume described here were designed to address this issue. Predictors of statistics of bed elevation fluctuations at short timescales, total and elevation‐specific particle entrainment rates, particle step lengths, mean and associated probability density function for particle virtual velocity, and thickness of the active layer were developed. The working hypothesis tested in this paper is that the statistics of tracer displacements can be related to channel‐averaged hydraulic parameters, and thus linked to macroscopic aspects of bed load transport.

One-Dimensional Modeling of Dam-Break Flow over Movable Beds

A one-dimensional model has been established to simulate the fluvial processes under dam-break flow over movable beds. The hydrodynamic model adopts the generalized shallow water equations, which consider the effects of sediment transport and bed change on the flow. The sediment model computes the nonequilibrium transport of bed load and suspended load. The effects of sediment concentration on sediment settling and entrainment are considered in determining the sediment settling velocity and transport capacity. In particular, a correction factor is proposed to modify the Van Rijn formulas of equilibrium bed-load transport rate and near-bed suspended-load concentration for the simulation of sediment transport under high-shear flow conditions. The governing equations are solved by an explicit finite-volume method with the first-order upwind scheme for intercell fluxes. The model has been tested in two experimental cases, with fairly good agreement between simulations and measurements. The sensitivities of the model results to parameters such as the sediment nonequilibrium adaptation length, Manning's roughness coefficient and the proposed correction factor have been verified. The proposed model has also been compared to an existing model and the results indicate the new model is more reliable.

Bed load at low Shields stress on arbitrarily sloping beds: Alternative entrainment formulation

According to the Bagnold hypothesis for equilibrium bed load transport, a necessary constraint for the maintenance of equilibrium bed load transport is that the fluid shear stress at the bed must be reduced to the critical, or threshold, value associated with incipient motion of grains. It was shown in a companion paper [Seminara et al., 2002], however, that the Bagnold hypothesis breaks down when applied to equilibrium bed load transport on beds with transverse slopes above a relatively modest value that is well below the angle of repose. An investigation of this failure resulted in a demonstration of its lack of validity even for nearly horizontal beds. The constraint is here replaced with an entrainment formulation, according to which a dynamic equilibrium is maintained by a balance between entrainment of bed grains into the bed load layer and deposition of bed load grains onto the bed. The entrainment function is formulated so that the entrainment rate is an increasing function of the excess of the fluid shear stress at the bed over the threshold value. The formulation is implemented with the aid of a unique set of laboratory data that characterizes equilibrium bed load transport at relatively low shear stresses for streamwise angles of bed inclination varying from nearly 0° to 22°. The formulation is shown to provide a description of bed load transport on nearly horizontal beds that fits the data as well as that resulting from the Bagnold constraint. The entrainment formulation has the added advantage of not requiring the unrealistically high dynamic coefficient of Coulomb friction resulting from the Bagnold constraint. Finally, the entrainment formulation provides reasonable and consistent results on finite streamwise and transverse bed slopes, even those at which the Bagnold formulation breaks down completely.

Bed load at low Shields stress on arbitrarily sloping beds: Failure of the Bagnold hypothesis

The Bagnold hypothesis has been a major tool in the development of mechanistic models of bed load transport. According to the hypothesis, a necessary constraint for the maintenance of equilibrium bed load transport is that the fluid shear stress at the bed must be reduced to the critical, or threshold value associated with incipient motion of grains. The constraint determines a functional form for the areal concentration of bed load grains as a function of flow parameters. It is shown here, however, that the Bagnold hypothesis breaks down when applied to equilibrium bed load transport on beds with transverse slopes above a relatively modest value that is well below the angle of repose. In particular, under such conditions it is shown that no areal concentration, no matter how large, is sufficient to reduce the fluid shear stress at the bed to the critical value. This failure motivates the abandonment of the Bagnold constraint, even for nearly horizontal beds. The framework presented here, however, provides a natural basis for an entrainment‐based model of sediment transport that neither satisfies nor suffers from the drawbacks of the Bagnold constraint.