Shear velocity

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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 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[edit]

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[edit]

  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