Law of the wall ($U/u_{\ast }={\kappa }^{-1}\log (z/z_0)$)

In a neutral turbulent constant-stress region, the stress velocity and wall distance give mean shear $U_z=u_{\ast }/(\kappa z)$. Its logarithmic integral uses the roughness length. The law applies outside the inner sublayer and below the outer boundary-layer scale, not all the way to a smooth wall.

The dimensionless coefficient in the logarithmic wall law and its mixing-length form $\ell =\kappa z$. Its numerical value is determined from the turbulence closure and measurements, rather than from dimensional analysis alone.

The extrapolation height at which a logarithmic wall profile vanishes. For a fully rough wall it is set mainly by the geometry; for a smooth wall it scales with $\nu /u_{\ast }$. It is not the thickness of the molecular viscous sublayer or the literal location of a no-slip surface.