# Shear velocity

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**Shear Velocity**, also called **friction velocity**, is a form by which a shear stress may be re-written in units of velocity. It is useful as a method in fluid mechanics to compare true velocities, such as the velocity of a flow in a stream, to a velocity that relates shear between layers of flow.

Shear velocity is used to describe shear-related motion in moving fluids. It is used to describe:

- Diffusion and dispersion of particles, tracers, and contaminants in fluid flows
- The velocity profile near the boundary of a flow (see Law of the wall)
- Transport of sediment in a channel

Shear velocity also helps in thinking about the rate of shear and dispersion in a flow. Shear velocity scales well to rates of dispersion and bedload sediment transport. A general rule is that the shear velocity is between 5% to 10% of the mean flow velocity.

For river base case, the shear velocity can be calculated by Manning's equation.

*n*is the Gauckler–Manning coefficient. Units for values of n are often left off, however it is not dimensionless, having units of: (T/[L^{1/3}]; s/[ft^{1/3}]; s/[m^{1/3}]).*R*is the hydraulic radius (L; ft, m);_{h}- the role of a is a dimension correct factor. Thus a= 1 m
^{1/3}/s = 1.49 ft^{1/3}/s.

Instead of finding and for your specific river of interest, you can examine the range of possible values and note that for most rivers, is between 5% and 10% of :

For general case

where *τ* is the shear stress in an arbitrary layer of fluid and *ρ* is the density of the fluid.

Typically, for sediment transport applications, the shear velocity is evaluated at the lower boundary of an open channel:

where *τ _{b}* is the shear stress given at the boundary.

Shear velocity can also be defined in terms of the local velocity and shear stress fields (as opposed to whole-channel values, as given above).

## Friction Velocity in Turbulence[edit]

The friction velocity is often used as a scaling parameter for the fluctuating component of velocity in turbulent flows.^{[1]} One method of obtaining the shear velocity is through non-dimensionalization of the turbulent equations of motion. For example, in a fully developed turbulent channel flow or turbulent boundary layer, the streamwise momentum equation in the very near wall region reduces to:

- .

By integrating in the *y*-direction once, then non-dimensionalizing with an unknown velocity scale *u*_{∗} and viscous length scale *ν*/*u*_{∗}, the equation reduces down to:

or

- .

Since the right hand side is in non-dimensional variables, they must be of order 1. This results in the left hand side also being of order one, which in turn give us a velocity scale for the turbulent fluctuations (as seen above):

- .

Here, *τ _{w}* refers to the local shear stress at the wall.

## Planetary boundary layer[edit]

Within the lowest portion of the planetary boundary layer a semi-empirical log wind profile is commonly used to describe the vertical distribution of horizontal mean wind speeds. The simplified equation that describe it is

where is the Von Kármán constant (~0.41), is the zero plane displacement (in metres).

The zero-plane displacement () is the height in meters above the ground at which zero wind speed is achieved as a result of flow obstacles such as trees or buildings. It can be approximated as ^{2}/_{3} to ^{3}/_{4} of the average height of the obstacles.^{[2]} For example, if estimating winds over a forest canopy of height 30 m, the zero-plane displacement could be estimated as d = 20 m.

Thus, you can extract the friction velocity by knowing the wind velocity at two levels (z).

## References[edit]

**^**Schlichting, H.; Gersten, K.*Boundary-Layer Theory*(8th ed.). Springer 1999. ISBN 978-81-8128-121-0.**^**Holmes JD. Wind Loading of Structures. 3rd ed. Boca Raton, Florida: CRC Press; 2015.

- Whipple, K. X. (2004). "III: Flow Around Bends: Meander Evolution" (PDF).
*MIT*. 12.163 Course Notes.