Abstract
The need for operational models describing the friction factor f in streams remains undisputed given its utility across a plethora of hydrological and hydraulic applications concerned with shallow inertial flows. For small-scale roughness elements uniformly covering the wetted parameter of a wide channel, the Darcy-Weisbach f = 8(u*/Ub)2 is widely used at very high Reynolds numbers, where u* is friction velocity related to the surface kinematic stress, Ub = Q/A is bulk velocity, Q is flow rate, and A is cross-sectional area orthogonal to the flow direction. In natural streams, the presence of vegetation introduces additional complications to quantifying f, the subject of the present work. Turbulent flow through vegetation are characterized by a number of coherent vortical structures: (i) von Karman vortex streets in the lower layers of vegetated canopies, (ii) Kelvin-Helmholtz as well as attached eddies near the vegetation top, and (iii) attached eddies well above the vegetated layer. These vortical structures govern the canonical mixing lengths for momentum transfer and their influence on f is to be derived. The main novelty is that the friction factor of vegetated flow can be expressed as fv = 4Cd(Uv/Ub)2 where Uv is the spatially averaged velocity within the canopy volume, and Cd is a local drag coefficient per unit frontal area derived to include the aforemontioned layer-wise effects of vortical structures within and above the canopy along with key vegetation properties. The proposed expression is compared with a number of empirical relations derived for vegetation under emergent and submerged conditions as well as numerous data sets covering a wide range of canopy morphology, densities, and rigidity. It is envisaged that the proposed formulation be imminently employed in eco-hydraulics where the interaction between flow and vegetation is being sought.
Highlights
To progress on the description of fv for canopy flows, a number of studies have been conducted to explore connections between the shape of the mean velocity profile u(z) and its depth-integrated value defined as ∫ Ub ≈ 1 hw hw u(z)dz, (3)in wide rectangular channels, where z is the distance from the channel bottom[23,24,25,26,27,28,29,30]
The work of Moody, Nikuradse and many others established f to vary with two dimensionless quantities: the relative roughness r/Rh and a bulk Reynolds number Reb,h = UbRh/ν, where ν is the kinematic viscosity of water. This expression is generally accepted in pipe- and open channel- flows above small-scale roughness elements where r/hw 1
We propose a 3-parameter mathematical function to describe this behavior without focusing on the detailed mean velocity profile in each zone
Summary
Wei-Jie Wang[1,2], Wen-Qi Peng[1,2], Wen-Xin Huai[3], Gabriel G. The work of Moody, Nikuradse and many others established f to vary with two dimensionless quantities: the relative roughness r/Rh and a bulk Reynolds number Reb,h = UbRh/ν, where ν is the kinematic viscosity of water This expression is generally accepted in pipe- and open channel- flows above small-scale roughness elements where r/hw 1. An immediate consequence of this result is that when r is a priori known, f and subsequently Sf can be determined from Eq 1 Such an expression for Sf can be used to mathematically close the combined continuity and unsteady shallow water flow equations (i.e., the Saint-Venant) to predict hw and Ub in a plethora of hydrological and hydraulics applications. It is shown that under certain simplifying assumptions, the canopy-related f (hereafter referred to as fv) is given by fv
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