# Shear velocity

**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 about ^{1}⁄_{10} of the mean flow velocity.

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

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.

## References

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