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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∗=⟨u⟩na(gRh−1/3)0.5{displaystyle u^{*}=langle urangle {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 correct factor. Thus a= 1 m^1/3/s = 1.49 ft^1/3/s.

Instead of finding n{displaystyle n} and Rh{displaystyle R_{h}} for your specific river of interest, you can examine the range of possible values and note that for most rivers, u∗{displaystyle u^{*}} is between 5% and 10% of ⟨u⟩{displaystyle langle urangle }:

For general case

u⋆=τρ{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:

u⋆=τbρ{displaystyle 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=ν∂2u¯∂y2−∂∂y(u′v′¯){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:

τwρ=ν∂u∂y−u′v′¯{displaystyle {frac {tau _{w}}{rho }}=nu {frac {partial u}{partial y}}-{overline {u'v'}}}

or

τwρu⋆2=∂u+∂y++τT+¯{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):

u⋆=τwρ{displaystyle 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)=u∗κ[ln⁡(z−dz0)]{displaystyle u(z)={frac {u_{*}}{kappa }}left[ln left({frac {z-d}{z_{0}}}right)right]}

where κ{displaystyle kappa } is the Von Kármán constant (~0.41), d{displaystyle d} is the zero plane displacement (in metres).

The zero-plane displacement (d{displaystyle 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∗=κ(u(z2)−u(z1))ln⁡(z2−dz1−d){displaystyle u_{*}={frac {kappa (u(z2)-u(z1))}{ln left({frac {z2-d}{z1-d}}right)}}}

References

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