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

This review describes mean and turbulent flow and mass transport in the presence of aquatic vegetation. Within emergent canopies, the turbulent length scales are set by the stem diameter and spacing, and the mean flow is determined by the distribution of the canopy frontal area. Near sparse submerged canopies, the bed roughness and near-bed turbulence are enhanced, but the velocity profile remains logarithmic. For dense submerged canopies, the drag discontinuity at the top of the canopy generates a shear layer, which contains canopy-scale vortices that control the exchange of mass and momentum between the canopy and the overflow. The canopy-scale vortices penetrate a finite distance into the canopy, δe, set by the canopy drag. This length scale segregates the canopy into two regions: The upper canopy experiences energetic turbulent transport, controlled by canopy-scale vortices, whereas the lower canopy experiences diminished transport, associated with the smaller stem-scale turbulence. The canopy-scale vortices induce a waving motion in flexible blades, called a monami.

Flow and Transport in Regions with Aquatic Vegetation

This paper highlights some recent trends in vegetation hydrodynamics, focusing on conditions within channels and spanning spatial scales from individual blades, to canopies or vegetation patches, to the channel reach. At the blade scale, the boundary layer formed on the plant surface plays a role in controlling nutrient uptake. Flow resistance and light availability are also influenced by the reconfiguration of flexible blades. At the canopy scale, there are two flow regimes. For sparse canopies, the flow resembles a rough boundary layer. For dense canopies, the flow resembles a mixing layer. At the reach scale, flow resistance is more closely connected to the patch-scale vegetation distribution, described by the blockage factor, than to the geometry of individual plants. The impact of vegetation distribution on sediment movement is discussed, with attention being paid to methods for estimating bed stress within regions of vegetation. The key research challenges of the hydrodynamics of vegetated channels ...

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Hydrodynamics of vegetated channels

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Formulas For Natural Frequency And Mode Shape

This paper summarizes recent advances in vegetation hydrodynamics and uses the new concepts to explore not only how vegetation impacts flow and transport, but also how flow feedbacks can influence vegetation spatial structure. Sparse and dense submerged canopies are defined based on the relative contribution of turbulent stress and canopy drag to the momentum balance. In sparse canopies turbulent stress remains elevated within the canopy and suspended sediment concentration is comparable to that in unvegetated regions. In dense canopies turbulent stress is reduced by canopy drag and suspended sediment concentration is also reduced. Further, for dense canopies, the length-scale of turbulence penetration into the canopy, δe, is shown to predict both the roughness height and the displacement height of the overflow profile. In a second case study, the relation between flow speed and spatial structure of a seagrass meadow gives insight into the stability of different spatial structures, defined by the area fraction covered by vegetation. In the last case study, a momentum balance suggests that in natural channels the total resistance is set predominantly by the area fraction occupied by vegetation, called the blockage factor, with little direct dependence on the specific canopy morphology.

/pdf/interaction-between-flow-transport-and-vegetation-spatial-3my09vbp5n.pdf

Interaction between flow, transport and vegetation spatial structure

The flow at the leading edge and in the interior of a rectangular porous obstruction is described through experiments and scaling. The porous obstruction consists of an emergent, rectangular array of cylinders in shallow flow, a configuration that mimics aquatic vegetation. The main features of the flow depend upon the non-dimensional canopy flow-blockage, which is a function of the obstruction width and porosity. For the ranges of canopy flow-blockage tested in this paper, the fluid decelerates upstream of the obstruction over a length scale proportional to the array width. For high flow-blockage, the interior adjustment length within the porous obstruction is set by the array width. For low flow-blockage, the array's frontal area per unit volume sets the interior adjustment length. Downstream of the adjustment regions, the interior velocity is governed by a balance between the lateral divergence of the turbulent stress and canopy drag, or by a balance between the pressure gradient and canopy drag, depending on the lateral penetration into the array of Kelvin–Helmholtz (KH) vortices, which is set by the non-dimensional canopy flow-blockage. For a porous obstruction with two stream-parallel edges, the KH vortex streets along the two edges are in communication across the width of the array: a phenomenon that results in cross-array vortex organization, which significantly enhances the vortex strength and creates significant lateral transport within the porous obstruction.

/pdf/flow-adjustment-and-interior-flow-associated-with-a-5e8qg4lpip.pdf

Flow adjustment and interior flow associated with a rectangular porous obstruction

Vegetation was added to a fully developed sandy point bar in the meander of a constructed stream. Significant changes in the flow structure and bed topography were observed. As expected, the addition of vegetative resistance decreased the depth-averaged streamwise velocity over the bar and increased it in the open region. In addition, the secondary circulation increased in strength but became confined to the deepest section of the channel. Over the point bar, the secondary flow was entirely outward, i.e., toward the outer bank. The changes in flow led to changes in bar shape. Although the region of the bar closest to the inner bank accumulated sediment, erosion of the bar and the removal of plants by scouring were observed at the interface between the planted bar and the open channel.

/pdf/effects-of-added-vegetation-on-sand-bar-stability-and-stream-27labv3fn7.pdf

Effects of Added Vegetation on Sand Bar Stability and Stream Hydrodynamics

We consider how the flapping of kelp blades may enhance the flux of nutrients to a blade, by stripping away the diffusive sub-layer and renewing the fluid at the blade surface. The surface renewal model explains the degree of flux enhancement observed in previous studies under different flow and flapping conditions. We measured the motion of real kelp blades of Laminaria saccharina, Macrocystis pyrifera, and Nereocystis luetkeana under unidirectional current in a laboratory flume. Observed flapping frequencies coupled with the renewal model, suggest that the flapping of blades in the field has the potential to significantly enhance flux to the blade surface at low current speed, but has little effect on flux at high current speeds.

Jeffrey Tsaros Rominger

Papers

Interaction between flow, transport and vegetation spatial structure

Flow adjustment and interior flow associated with a rectangular porous obstruction

Effects of Added Vegetation on Sand Bar Stability and Stream Hydrodynamics

Effects of Added Vegetation on Sand Bar Stability and Stream Hydrodynamics

The motion of kelp blades and the surface renewal model