# 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.

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

Where $\tau$ is the shear stress in an arbitrary layer of fluid and $\rho$ is the density of the fluid.

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

$u_{\star}=\sqrt{\frac{\tau_b}{\rho}}$

Where $\tau_b$ 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 stream-wise momentum equation in the very near wall region reduces to:

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

By integrating in the y-direction once, then non-dimensionalizing with an unknown velocity scale $u_{\star}$ and viscous length scale $\frac{\nu}{u_{\star}}$, the equation reduces down to:

$\frac{\tau_w}{\rho} = \nu\frac{\partial u}{\partial y} - \overline{u'v'}$

or

$\frac{\tau_w}{\rho u_{\star}^2} = \frac{\partial u^+}{\partial y^+} + \overline{\tau_T^+}$.

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):

$u_{\star} = \sqrt{\frac{\tau_w}{\rho}}$.

Here, $\tau_w$ refers to the local shear stress at the wall.

## References

1. ^ H. Schlichting, K. Gersten, "Boundary-Layer Theory" 8th edition Springer 1999 ISBN 978-81-8128-121-0.

Whipple, K. X (2004), III: Flow Around Bends: Meander Evolution, 12.163 Course Notes, MIT. http://ocw.mit.edu/courses/earth-atmospheric-and-planetary-sciences/12-163-surface-processes-and-landscape-evolution-fall-2004/lecture-notes/3_flow_around_bends.pdf