Shear stress velocity

The shear velocity is in the hydrodynamics a measure of the shear stress , a layer of a flowing fluid on an adjacent layer or a boundary surface exerts. It is calculated from the amount of the shear stress vector and the density of the fluid: ${\displaystyle u_{*}}$ ${\displaystyle \tau }$ ${\displaystyle {\vec {\tau }}={\begin{pmatrix}\tau _{x}\\\tau _{y}\end{pmatrix}}}$ ${\displaystyle \rho }$

${\displaystyle u_{*}={\sqrt {\tau /\rho }}}$

In turbulent media , the shear stress is dominated by the turbulent transport. The components of the shear stress vector are then calculated from the elements of Reynolds' shear stress tensor :

${\displaystyle {\begin{aligned}\tau _{x}&=-\rho \cdot {\overline {u'w'}}\\\tau _{y}&=-\rho \cdot {\overline {v'w'}}\end{aligned}}}$

in which

• ${\displaystyle u}$and the two velocity components parallel (x- and y-direction) and perpendicular (z-direction) to the interface as well ${\displaystyle v}$ ${\displaystyle w}$
• canceled quantities such as the deviations from the mean. ${\displaystyle u'=u-{\overline {u}}}$

The two covariances and can also be interpreted as the turbulent flows in the z direction of the momentum in the x or y direction. ${\displaystyle \tau _{x}}$ ${\displaystyle \tau _{y}}$

The shear stress velocity is the square root of the amount of this vector:

${\displaystyle u_{*}={\sqrt {|{\vec {\tau }}|/\rho }}={\sqrt {{\sqrt {\tau _{x}^{2}+\tau _{y}^{2}}}/\rho }}={\sqrt {\sqrt {{\overline {u'w'}}^{2}+{\overline {v'w'}}^{2}}}}}$

When deriving the logarithmic wind profile using the mixing path length approach according to Ludwig Prandtl , the coordinate system is defined so that the x-axis is parallel to the mean wind direction ( ), and it is assumed that the mean wind direction and the direction of the shear stress coincide ( ). In this case: ${\displaystyle {\overline {v}}=0}$ ${\displaystyle \tau _{y}=0}$

${\displaystyle u_{*}^{2}=-{\overline {w'u'}}}$

See also

Wall shear stress velocity

literature

• Erich Truckenbrodt: Fundamentals and elementary flow processes of density-stable fluids . In: Fluid Mechanics . 4th edition. tape 1 . Springer-Verlag, Berlin / Heidelberg 1996, ISBN 978-3-540-79017-4 .