Consider Prandtl's mixing length hypothesis

(18.15)

Where, the mixing length is given by

(18.16)

The parameter is called Von-Karman constant.

Here, Karman assumed that should be a function of the distance from the wall in a wall bounded turbulent flow, rather than a constant as taken by Prandtl in the case of free turbulent flows. Substituting (18.15) and (18.16) into (18.14), we obtain

	(18.17)
	(18.18)
	(18.19)

Integrating (18.19) between 0 and y⁺ , we get

(18.20)

where, and . These are near-universal constants for turbulent flow past smooth impermeable walls eventhough C varies slightly with the pressure gradient.

Equation (18.20) is called the Logarithmic Law of the wall, which is valid for , the so called inertial sublayer.

Barenblatt and Chorin (1998) introduced a Law of the wall which is given by

(18.21)

where and

The law emphasizes the dependence of the equation on the Reynolds number.

Reference