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% to 10% of the mean flow velocity.

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

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 $n$ and $R_{h}$ for your specific river of interest, you can examine the range of possible values and note that for most rivers, $u^{*}$ is between 5% and 10% of $\langle u\rangle$ :

For general case

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:

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

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:

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:

{\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, τ_w refers to the local shear stress at the wall.

Planetary boundary layer

Within the lowest portion of the planetary boundary layer a semi-empirical log wind profile is commonly used to describe the vertical distribution of horizontal mean wind speeds. The simplified equation that describe it is

$u(z)={\frac {u_{*}}{\kappa }}\left[\ln \left({\frac {z-d}{z_{0}}}\right)\right]$

where $\kappa$ is the Von Kármán constant (~0.41), $d$ is the zero plane displacement (in metres).

The zero-plane displacement ( $d$ ) is the height in meters above the ground at which zero wind speed is achieved as a result of flow obstacles such as trees or buildings. It can be approximated as ²/₃ to ³/₄ of the average height of the obstacles.[2] For example, if estimating winds over a forest canopy of height 30 m, the zero-plane displacement could be estimated as d = 20 m.

Thus, you can extract the friction velocity by knowing the wind velocity at two levels (z).

$u_{*}={\frac {\kappa (u(z2)-u(z1))}{\ln \left({\frac {z2-d}{z1-d}}\right)}}$

References

Schlichting, H.; Gersten, K. Boundary-Layer Theory (8th ed.). Springer 1999. ISBN 978-81-8128-121-0.
Holmes JD. Wind Loading of Structures. 3rd ed. Boca Raton, Florida: CRC Press; 2015.

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

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

[Holmes2015-2] Holmes JD. Wind Loading of Structures. 3rd ed. Boca Raton, Florida: CRC Press; 2015.