Wall bounded turbulent shear flow: Analytic result for an universal amplitude

In the turbulent boundary layer above a flat plate, the velocity profile is known to have the form v=v_0[(1/\kappa) ln z + constant]. The distance from the wall in dimensionless units is z and v_0 is an uniquely defined velocity scale. The number \kappa is universal and measurements over several decades have shown that it is nearly 0.42. We use a randomly stirred model of turbulence to derive the above law and find \kappa=\sqrt{\frac{108} {125 \pi}} \simeq 0.52.