User:Srw2023/Sediment transport

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Settling velocity

Streamlines around a sphere falling through a fluid. This illustration is accurate for laminar flow, in which the particle Reynolds number is small. This is typical for small particles falling through a viscous fluid; larger particles would result in the creation of a turbulent wake.

The settling velocity (also called the "fall velocity" or "terminal velocity") is a function of the particle Reynolds number. Generally, for small particles (laminar approximation), it can be calculated with Stokes' Law. For larger particles (turbulent particle Reynolds numbers), fall velocity is calculated with the turbulent drag law. Dietrich (1982) compiled a large amount of published data to which he empirically fit settling velocity curves. Ferguson and Church (2006) analytically combined the expressions for Stokes flow and a turbulent drag law into a single equation that works for all sizes of sediment, and successfully tested it against the data of Dietrich. Their equation is

$w_{s}={\frac {RgD^{2}}{C_{1}\nu +(0.75C_{2}RgD^{3})^{(0.5)}}}$.

In this equation w_s is the sediment settling velocity, g is acceleration due to gravity, and D is mean sediment diameter. $\nu$ is the kinematic viscosity of water, which is approximately 1.0 x 10⁻⁶ m²/s for water at 20 °C.

$C_{1}$ and $C_{2}$ are constants related to the shape and smoothness of the grains.

Constant	Smooth Spheres	Natural Grains: Sieve Diameters	Natural Grains: Nominal Diameters	Limit for Ultra-Angular Grains
$C_{1}$	18	18	20	24
$C_{2}$	0.4	1.0	1.1	1.2

The expression for fall velocity can be simplified so that it can be solved only in terms of D. We use the sieve diameters for natural grains, $g=9.8$, and values given above for $\nu$ and $R$. From these parameters, the fall velocity is given by the expression:

$w_{s}={\frac {16.17D^{2}}{1.8\cdot 10^{-5}+(12.1275D^{3})^{(0.5)}}}$

Alternatively, settling velocity for a particle of sediment can be also be derived using Stokes Law assuming quiescent (or still) fluid in steady state. The resulting formulation for settling velocity is,

$w_{s}={\frac {g~({\frac {\rho _{s}-\rho }{\rho }})~d_{sed}^{2}}{18\nu }}$,

where $g$ is the gravitational constant, 9.81 $m/s^{2}$; $\rho _{s}$ is the density of the sediment; $\rho$ is the density of water; $d_{sed}$ is the sediment particle diameter (commonly assumed to be the median particle diameter, often referred to as $d_{50}$ in field studies); and $\nu$ is the molecular viscosity of water. The Stokes settling velocity can be thought of the terminal velocity resulting from balancing a particles' buoyant (proportional to the cross-sectional area) and gravitational forces (proportional to the mass). Small particles will have a slower settling velocity than heavier particles, as seen in the figure. This has implications for many aspects of sediment transport, for example, how far downstream a particle might be advected in a river.  

A plot showing the relationship between sediment particle diameter and the Stokes settling velocity