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 between 5% and 10% of the mean flow velocity.

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

${\displaystyle u^{*}=\langle u\rangle {\frac {n}{a}}(gR_{h}^{-1/3})^{0.5}}$

• 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_h is the hydraulic radius (L; ft, m);
• the role of a is a dimension correction factor. Thus a= 1 m^1/3/s = 1.49 ft^1/3/s.

Instead of finding ${\displaystyle n}$ and ${\displaystyle R_{h}}$ for the specific river of interest, the range of possible values can be examined; for most rivers, ${\displaystyle u^{*}}$ is between 5% and 10% of ${\displaystyle \langle u\rangle }$:

For general case

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

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 τ_b is the shear stress given at the boundary.

Shear velocity is linked to the Darcy friction factor by equating wall shear stress, giving:

${\displaystyle u_{\star }={\langle u\rangle }{\sqrt {\frac {f_{\mathrm {D} }}{8}}}}$

where f_D is the friction factor.

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).