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

The complexity of flow and wide variety of depositional processes operating in subaqueous density flows, combined with post-depositional consolidation and soft-sediment deformation, often make it difficult to interpret the characteristics of the original flow from the sedimentary record. This has led to considerable confusion of nomenclature in the literature. This paper attempts to clarify this situation by presenting a simple classification of sedimentary density flows, based on physical flow properties and grain-support mechanisms, and briefly discusses the likely characteristics of the deposited sediments. Cohesive flows are commonly referred to as debris flows and mud flows and defined on the basis of sediment characteristics. The boundary between cohesive and non-cohesive density flows (frictional flows) is poorly constrained, but dimensionless numbers may be of use to define flow thresholds. Frictional flows include a continuous series from sediment slides to turbidity currents. Subdivision of these flows is made on the basis of the dominant particle-support mechanisms, which include matrix strength (in cohesive flows), buoyancy, pore pressure, grain-to-grain interaction (causing dispersive pressure), Reynolds stresses (turbulence) and bed support (particles moved on the stationary bed). The dominant particle-support mechanism depends upon flow conditions, particle concentration, grain-size distribution and particle type. In hyperconcentrated density flows, very high sediment concentrations (>25 volume%) make particle interactions of major importance. The difference between hyperconcentrated density flows and cohesive flows is that the former are friction dominated. With decreasing sediment concentration, vertical particle sorting can result from differential settling, and flows in which this can occur are termed concentrated density flows. The boundary between hyperconcentrated and concentrated density flows is defined by a change in particle behaviour, such that denser or larger grains are no longer fully supported by grain interaction, thus allowing coarse-grain tail (or dense-grain tail) normal grading. The concentration at which this change occurs depends on particle size, sorting, composition and relative density, so that a single threshold concentration cannot be defined. Concentrated density flows may be highly erosive and subsequently deposit complete or incomplete Lowe and Bouma sequences. Conversely, hydroplaning at the base of debris flows, and possibly also in some hyperconcentrated flows, may reduce the fluid drag, thus allowing high flow velocities while preventing large-scale erosion. Flows with concentrations <9% by volume are true turbidity flows (sensuBagnold, 1962), in which fluid turbulence is the main particle-support mechanism. Turbidity flows and concentrated density flows can be subdivided on the basis of flow duration into instantaneous surges, longer duration surge-like flows and quasi-steady currents. Flow duration is shown to control the nature of the resulting deposits. Surge-like turbidity currents tend to produce classical Bouma sequences, whose nature at any one site depends on factors such as flow size, sediment type and proximity to source. In contrast, quasi-steady turbidity currents, generated by hyperpycnal river effluent, can deposit coarsening-up units capped by fining-up units (because of waxing and waning conditions respectively) and may also include thick units of uniform character (resulting from prolonged periods of near-steady conditions). Any flow type may progressively change character along the transport path, with transformation primarily resulting from reductions in sediment concentration through progressive entrainment of surrounding fluid and/or sediment deposition. The rate of fluid entrainment, and consequently flow transformation, is dependent on factors including slope gradient, lateral confinement, bed roughness, flow thickness and water depth. Flows with high and low sediment concentrations may co-exist in one transport event because of downflow transformations, flow stratification or shear layer development of the mixing interface with the overlying water (mixing cloud formation). Deposits of an individual flow event at one site may therefore form from a succession of different flow types, and this introduces considerable complexity into classifying the flow event or component flow types from the deposits.

The physical character of subaqueous sedimentary density flows and their deposits

応用水文 = Applied hydrology

The dynamics and stability of thin liquid films have fascinated scientists over many decades: the observations of regular wave patterns in film flows down a windowpane or along guttering, the patterning of dewetting droplets, and the fingering of viscous flows down a slope are all examples that are familiar in daily life. Thin film flows occur over a wide range of length scales and are central to numerous areas of engineering, geophysics, and biophysics; these include nanofluidics and microfluidics, coating flows, intensive processing, lava flows, dynamics of continental ice sheets, tear-film rupture, and surfactant replacement therapy. These flows have attracted considerable attention in the literature, which have resulted in many significant developments in experimental, analytical, and numerical research in this area. These include advances in understanding dewetting, thermocapillary- and surfactant-driven films, falling films and films flowing over structured, compliant, and rapidly rotating substrates, and evaporating films as well as those manipulated via use of electric fields to produce nanoscale patterns. These developments are reviewed in this paper and open problems and exciting research avenues in this thriving area of fluid mechanics are also highlighted.

Dynamics and stability of thin liquid films

Aquatic plants convert mean kinetic energy into turbulent kinetic energy at the scale of the plant stems and branches. This energy transfer, linked to wake generation, affects vegetative drag and turbulence intensity. Drawing on this physical link, a model is developed to describe the drag, turbulence and diffusion for flow through emergent vegetation which for the first time captures the relevant underlying physics, and covers the natural range of vegetation density and stem Reynolds' numbers. The model is supported by laboratory and field observations. In addition, this work extends the cylinder-based model for vegetative resistance by including the dependence of the drag coefficient, CD, on the stem population density, and introduces the importance of mechanical diffusion in vegetated flows.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/1998WR900069

Drag, turbulence, and diffusion in flow through emergent vegetation

Sedimentation Engineering: Theories, Measurements, Modeling, and Practice (ASCE Manuals and Reports on Engineering Practice No. 110) is intended to supplement ""Sedimentation Engineering: Classic Edition"" (ASCE Manuals and Reports on Engineering Practice No. 54), a seminal text on the nature and scope of sedimentation problems, methods for their investigation, and practical approaches to their solution.""Sedimentation Engineering: Theories, Measurements, Modeling, and Practice"" updates selected topics in the original manual and addresses new topics, such as the hydraulic action of flowing water in the context of rivers and inland water bodies.

Sedimentation engineering : processes, measurements, modeling, and practice

The literature abounds with apparently conflicting relations for predicting the reference near-bed concentration of suspended sediment or the entrainment rate of noncohesive sediment into suspension. Most of these relations have been developed for uniform material. In this paper, seven such relations are compared against a common set of experimental data for which direct measurements of near-bed concentration are available. The relations put forward by Van Rijn and by Smith and McLean are found to perform best. A new relation for uniform material, having similar predictive capabilities, is developed using the same experimental data. The new relation is generalized to handle sediment mixtures with the aid of field data. The generalization accounts for the hiding effect due to the presence of grains of different sizes. A test against an independent field data set indicates that the empirical fit can provide reasonable estimates of the sediment entrainment coefficient for beds covered with nonuniform material.

Entrainment of Bed Sediment into Suspension

The ability of turbulence models, based on two equation closure schemes (the k-e and the k-ω formulations) to compute the mean flow and turbulence structure in open channels with rigid, nonemergent vegetation is analyzed. The procedure, developed by Raupach and Shaw (1982), for atmospheric flows over plant canopies is used to transform the 3D problem into a more tractable 1D framework by averaging the conservation laws over space and time. With this methodology, form/drag related terms arise as a consequence of the averaging procedure, and do not need to be introduced artificially in the governing equations. This approach resolves the apparent ambiguity in previously reported values of the drag-related weighting coefficients in the equations for the turbulent kinetic energy and dissipation rates. The working hypothesis for the numerical models is that the flux gradient approximation applies to spatial/temporal averaged conservation laws, so that the eddy viscosity concept can be used. Numerical results ar...

Mean Flow and Turbulence Structure of Open-Channel Flow through Non-Emergent Vegetation

Experiments were conducted to determine the behaviour of turbidity currents laden with non-cohesive silt (silica flour) moving down a slope the bed of which is covered with similar silt. Each current was sustained with constant inlet conditions for six to eight minutes. The motion of the head was not studied; measurements were concentrated on the continuous part of the current that was essentially constant in time but developing in space. Only supercritical currents were studied. The currents were free to erode sediment from, and deposit sediment on, the bed. Measurements of vertical profiles of downstream velocity and sediment concentration allowed for the development of approximate similarity relations. These relations can be used to estimate most of the shape factors that occur in the vertically-integrated equations of motion. The values do not deviate grossly from those found by using a "top-hat" function, i.e. a step function, such that velocity or concentration equals its average value in the layer ...

Experiments on turbidity currents over an erodible bed

A high-speed video system was used to study the interaction between sediment particles and turbulence in the wall region of an open channel flow with both smooth and transitionally rough beds. In smooth flows, particles immersed within the viscous sublayer were seen to accumulate along low-speed wall streaks; apparently due to the presence of quasi-streamwise vortices in the wall region. Larger particles did not tend to group along streaks, however their velocity was observed to respond to the streaky structure of the flow velocity in the wall region. In transitionally rough flows particle sorting was not observed. Coherent flow structures in the form of shear layers typically observed in the near-wall region interacted with sediment particles lying on the channel bottom, resulting in the particles being entrained into suspension. Although there has been some speculation that this process would not be effective in entraining particles totally immersed in the viscous sublayer, the results obtained demonstrate the opposite. The entrainment mechanism appears to be the same independent of the roughness condition of the bottom wall, smooth or transitionally rough. In the latter case, however, hiding effects tend to preclude the entrainment of particles with sizes finer than that of the roughness elements. The analysis of particle velocity during entrainment shows that the streamwise component tends to be much smaller than the local mean flow velocity, while the vertical component tends to be much larger than the local standard deviation of the vertical flow velocity fluctuations, which would indicate that such particles are responding to rather extreme flow ejection events.

Marcelo H. Garcia

Papers

Sedimentation engineering : processes, measurements, modeling, and practice

Entrainment of Bed Sediment into Suspension

Mean Flow and Turbulence Structure of Open-Channel Flow through Non-Emergent Vegetation

Experiments on turbidity currents over an erodible bed

Experiments on particle—turbulence interactions in the near–wall region of an open channel flow: implications for sediment transport