The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Field data are used to study the size distribution of bedload in paved gravel-bed streams. Similarity analysis yields the results that all grain size ranges are of approximately equal transportability when the critical condition for breaking the pavement is exceeded. This result is only approximately correct due to deviations from similarity. However, it is adequate to justify development of a method for calculating total bedload, which requires only the subpavement median grain size rather than the size distribution. A method for calculating bedload size distribution that accounts for deviation from similarity is also developed.

Closure of "Bedload and Size Distribution in Paved Gravel-Bed Streams"

Geomorphology is currently in a period of resurgence as we seek to explain the diversity, origins and dynamics of terrain on the Earth and other planets in an era of increased environmental awareness. Yet there is a great deal we still do not know about the physics and chemistry of the processes that weaken rock and transport mass across a planet's surface. Discovering and refining the relevant geomorphic transport functions requires a combination of careful field measurements, lab experiments, and use of longer-term natural experiments to test current theory and develop new understandings. Landscape evolution models have an important role to play in sharpening our thinking, guiding us toward the right observables, and mapping out the logical consequences of transport laws, both alone and in combination with other salient processes. Improved quantitative characterization of terrain and process, and an ever-improving theory that describes the continual modification of topography by the many and varied processes that shape it, together with improved observation and qualitative and quantitative modelling of geology, vegetation and erosion processes, will provide insights into the mechanisms that control catchment form and function. This paper reviews landscape theory – in the form of numerical models of drainage basin evolution and the current knowledge gaps and future computing challenges that exist. Copyright © 2010 John Wiley & Sons, Ltd.

Modelling landscape evolution

Data from laboratory flumes and natural streams show that the critical Shields stress for initial sediment motion increases with channel slope, which indicates that particles of the same size are more stable on steeper slopes. This observation is contrary to standard models that predict reduced stability with increasing slope due to the added downstream gravitational force. Processes that might explain this discrepancy are explored using a simple force-balance model, including increased drag from channel walls and bed morphology, variable friction angles, grain emergence, flow aeration, and changes to the local flow velocity and turbulent fluctuations. Surprisingly, increased drag due to changes in bed morphology does not appear to be the cause of the slope dependency because both the magnitude and trend of the critical Shields stress are similar for flume experiments and natural streams, and significant variations in bed morphology in flumes is unlikely. Instead, grain emergence and changes in local flow velocity and turbulent fluctuations seem to be responsible for the slope dependency due to the coincident increase in the ratio of bed-roughness scale to flow depth (i.e., relative roughness). A model for the local velocity within the grain-roughness layer is proposed based on a 1-D eddy viscosity with wake mixing. In addition, the magnitude of near-bed turbulent fluctuations is shown to depend on the depth-averaged flow velocity and the relative roughness. Extension of the model to mixed grain sizes indicates that the coarser fraction becomes increasingly difficult to transport on steeper slopes.

/pdf/is-the-critical-shields-stress-for-incipient-sediment-motion-2at0gou7az.pdf

Is the critical Shields stress for incipient sediment motion dependent on channel‐bed slope?

The article surveys the current state of our understanding of turbidity currents, with an emphasis on their fluid mechanics. It highlights the significant role these currents play within the global sediment cycle, and their importance in environmental processes and in the formation of hydrocarbon reservoirs. Events and mechanisms governing the initiation of turbidity currents are reviewed, along with experimental observations and findings from field studies regarding their internal velocity and density structure. As turbidity currents propagate over the seafloor, they can trigger the evolution of a host of topographical features through the processes of deposition and erosion, such as channels, levees, and sediment waves. Potential linear instability mechanisms are discussed that may determine the spatial scales of these features. Finally, the hierarchy of available theoretical models for analyzing the dynamics of turbidity currents is outlined, ranging from dimensional analysis and integral models to both depth-averaged and depth-resolving simulation approaches.

Turbidity Currents and Their Deposits

Temporally and spatially averaged models of bedload transport are inadequate to describe the highly variable nature of particle motion at low transport stages. The primary sources of this variability are the resisting forces to downstream motion resulting from the geometrical relation (pocket friction angle) of a bed grain to the grains that it rests upon, variability of the near-bed turbulent velocity field and the local modification of this velocity field by upstream, protruding grains. A model of bedload transport is presented that captures these sources of variability by directly integrating the equations of motion of each particle of a simulated mixed grain-size sediment bed. Experimental data from the velocity field downstream and below the tops of upstream, protruding grains are presented. From these data, an empirical relation for the velocity modification resulting from upstream grains is provided to the bedload model. The temporal variability of near-bed turbulence is provided by a measured near-bed time series of velocity over a gravel bed. The distribution of pocket friction angles results as a consequence of directly calculating the initiation and cessation of motion of each particle as a result of the combination of fluid forcing and interaction with other particles. Calculations of bedload flux in a uniform boundary and simulated pocket friction angles agree favourably with previous studies.

Direct numerical simulation of bedload transport using a local, dynamic boundary condition

[1] In natural flows, bed sediment particles are entrained and moved by the fluctuating forces, such as lift and drag, exerted by the overlying flow on the particles To develop a better understanding of these forces and the relation of the forces to the local flow, the downstream and vertical components of force on near-bed fixed particles and of fluid velocity above or in front of them were measured synchronously at turbulence-resolving frequencies (200 or 500 Hz) in a laboratory flume Measurements were made for a spherical test particle fixed at various heights above a smooth bed, above a smooth bed downstream of a downstream-facing step, and in a gravel bed of similarly sized particles as well as for a cubical test particle and 7 natural particles above a smooth bed Horizontal force was well correlated with downstream velocity and not correlated with vertical velocity or vertical momentum flux The standard drag formula worked well to predict the horizontal force, but the required value of the drag coefficient was significantly higher than generally used to model bed load motion For the spheres, cubes, and natural particles, average drag coefficients were found to be 076, 136, and 091, respectively For comparison, the drag coefficient for a sphere settling in still water at similar particle Reynolds numbers is only about 04 The variability of the horizontal force relative to its mean was strongly increased by the presence of the step and the gravel bed Peak deviations were about 30% of the mean force for the sphere over the smooth bed, about twice the mean with the step, and 4 times it for the sphere protruding roughly half its diameter above the gravel bed Vertical force correlated poorly with downstream velocity, vertical velocity, and vertical momentum flux whether measured over or ahead of the test particle Typical formulas for shear-induced lift based on Bernoulli's principle poorly predict the vertical forces on near-bed particles The measurements suggest that particle-scale pressure variations associated with turbulence are significant in the particle momentum balance

https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/2006JF000536

Forces on stationary particles in near-bed turbulent flows

A model of sand transport in water is produced by combining a turbulence-resolving large eddy simulation (LES) with a discrete element model (DEM) prescribing the motion of individual grains of medium sand. The momentum effect of each particle on the fluid is calculated at the LES cell containing the particle, and the fluid velocity and pressure, interpolated to each particle center, is used to derive fluid force on each particle in the DEM. Eleven numerical experiments are conducted of an initially flat bed of particles. The experiments span a range of motion, from essentially no motion to vigorous suspension. Hydraulic roughness is found to increase abruptly at the transition from bed load to suspended load transport. Suspended sediment extracts momentum from the flow and decreases the rate of shear. Whereas, slightly higher in the flow, vertical drag by suspended grains damps turbulence and increases the rate of shear. Vertical sediment diffusivity and effective particle settling velocity are much smaller than is commonly assumed in suspended sediment models. The bed load experiments suggest that saltation by itself is a poor model of bed load sand transport. In contrast to expectations from saltation models, the peak bed load flux occurs at essentially the same level as the bed, and grains move slowly in frequent contact with other grains. Higher- and faster-moving bed load grains that can be considered to be in saltation represent a smaller portion of the total flux. Entrainment of bed load grains occurs in response to fluid penetration of the bed by high-vorticity turbulence structures embedded within broader high speed fluid regions referred to as a sweeps or high-speed wedges.

Numerical simulation of turbulence and sediment transport of medium sand

[1] We provide a probabilistic definition of the bed load sediment flux. In treating particle positions and motions as stochastic quantities, a flux form of the Master equation (a general expression of conservation) reveals that the volumetric flux involves an advective part equal to the product of an average particle velocity and the particle activity (the solid volume of particles in motion per unit streambed area), and a diffusive part involving the gradient of the product of the particle activity and a diffusivity that arises from the second moment of the probability density function of particle displacements. Gradients in the activity, instantaneous or time-averaged, therefore effect a particle flux. Time-averaged descriptions of the flux involve averaged products of the particle activity, the particle velocity and the diffusivity; the significance of these products depends on the scale of averaging. The flux form of the Exner equation looks like a Fokker-Planck equation (an advection-diffusion form of the Master equation). The entrainment form of the Exner equation similarly involves advective and diffusive terms, but because it is based on the joint probability density function of particle hop distances and associated travel times, this form involves a time derivative term that represents a lag effect associated with the exchange of particles between the static and active states. The formulation is consistent with experimental measurements and simulations of particle motions reported in companion papers.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2012JF002352

A probabilistic description of the bed load sediment flux: 1. Theory

[1] High-speed imaging of coarse sand particles transported as bed load over a planar bed reveals that the particle activity, the solid volume of particles in motion per unit streambed area, fluctuates as particles respond to near-bed fluid turbulence while simultaneously interacting with the bed. The relative magnitude of these fluctuations systematically varies with the size of the sampling area. The particle activity within a specified sampling area is distributed in a manner that is consistent with the existence of an ensemble of configurations of particle positions wherein certain configurations are preferentially selected or excluded by the turbulence structure, manifest as patchiness of active particles. The particle activity increases with increasing bed stress far faster than does the average particle velocity, so changes in the transport rate with changing stress are dominated by changes in the activity, not velocity. The probability density functions of the streamwise and cross-stream particle velocities are exponential-like and lack heavy tails. Plots of the mean squared particle displacement versus time may ostensibly indicate non-Fickian diffusive behavior while actually reflecting effects of correlated random walks associated with intrinsic periodicities in particle motions, not anomalous diffusion. The probability density functions of the particle hop distance (start-to-stop) and the associated travel time are gamma-like, which provides the empirical basis for showing that particle disentrainment rates vary with hop distance and travel time.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2012JF002353

Mark W. Schmeeckle

Papers

Direct numerical simulation of bedload transport using a local, dynamic boundary condition

Forces on stationary particles in near-bed turbulent flows

Numerical simulation of turbulence and sediment transport of medium sand

A probabilistic description of the bed load sediment flux: 1. Theory

A probabilistic description of the bed load sediment flux: 2. Particle activity and motions