# 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 110 of the mean flow velocity.

${\displaystyle u_{\star }={\sqrt {\frac {\tau }{\rho }}}}$

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:

${\displaystyle u_{\star }={\sqrt {\frac {\tau _{b}}{\rho }}}}$

where τb 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:

${\displaystyle 0={\nu }{\partial ^{2}{\overline {u}} \over \partial y^{2}}-{\frac {\partial }{\partial y}}({\overline {u'v'}})}$.

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:

${\displaystyle {\frac {\tau _{w}}{\rho }}=\nu {\frac {\partial u}{\partial y}}-{\overline {u'v'}}}$

or

${\displaystyle {\frac {\tau _{w}}{\rho u_{\star }^{2}}}={\frac {\partial u^{+}}{\partial y^{+}}}+{\overline {\tau _{T}^{+}}}}$.

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):

${\displaystyle u_{\star }={\sqrt {\frac {\tau _{w}}{\rho }}}}$.

Here, τw refers to the local shear stress at the wall.

## References

1. ^ Schlichting, H.; Gersten, K. Boundary-Layer Theory (8th ed.). Springer 1999. ISBN 978-81-8128-121-0.