The Navier – stokes equations describe the transport of momentum in a viscous fluid. For a laminar flow, these N.S.E. can be solved directly, often to a high degree of accuracy.
For turbulent flow, things are more complex. The equations describe the instantaneous velocity components u, v, and w at every point in the flow. However, the nature of turbulence is such that there are very strong variation in these quantities over small distance. The time over which fluctuations in velocity occurs are likewise very small.
In this chapter a brief explanation and derivation of the N.S.E. for turbulent flow which they called "Reynolds stress".
Prandtl mixing length theory also presented to solve the "Reynolds stress" related to a length scale and velocity gradient. In addition, the velocity profiles for turbulent flow described throughout an experimental variation of inner – outer and overlap layer laws.