Gibbs lemma

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.

$u_{⋆} = \sqrt{\frac{τ}{ρ}}$

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:

$u_{⋆} = \sqrt{\frac{τ_{b}}{ρ}}$

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).