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

TL;DR

This paper derives an analytic expression for the universal velocity profile in turbulent boundary layers, predicting a value of the von Kármán constant close to 0.52 using a randomly stirred turbulence model.

Contribution

It provides an analytic derivation of the universal velocity profile and the von Kármán constant in turbulent shear flows, improving theoretical understanding.

Findings

Derived the velocity profile law analytically.

Predicted the von Kármán constant as approximately 0.52.

Compared theoretical value with experimental measurements.

Abstract

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.