Reynolds Equation and Mixing Length Model

Turbulent Flow Between Two Parallel Plates

Logarithmic Friction Law

The velocity defect law and the law of the wall in the inertial sublayer are given as

(24)

(25)

Subtracting, we find

, ,

(26)

with and  known, equation (26) is the statement of the logarithmic friction law.

Pipe Flow

The law of the wall and the velocity defect law are also valid for turbulent pipe flows. Equations (9) - (26) can be written for pipe flows with the following minor changes:

, ,

(27)

Here,  is the radius of the pipe and  is the distance from the wall. For pipe flows,  and equations (24) - (26) become

, is valid up to   ,

(28)

,

(29)

.

(30)