Turbulent velocity profile

Turbulent velocity profile 
A typical comparison of laminar and turbulent velocity profiles for wall turbulent flows, are shown in Fig. 5.1.4(a-b). The nature of the profile is parabolic in the case of laminar flow and the same trend is seen in the case of turbulent flow at the wall. The typical measurements across a turbulent flow near the wall have three distinct zones as shown in Fig. 5.1.4(c). The outer layer is of two or three order magnitudes greater than the wall layer and vice versa. Hence, the different sub-layers of Eq. (5.1.11) may be defined as follows; 
•  Wall layer (laminar shear dominates) 
•  Outer layer (turbulent shear dominates) 
•  Overlap layer (both types of shear are important) 
Fig 5.1.4: Velocity and shear layer distribution: (a) velocity profile in laminar flow; (b) velocity profile in turbulent flow; (c) shear layer in a turbulent flow. 
In a typical turbulent flow, let the wall shear stress, thickness of outer layer and velocity at the edge of the outer layer be , respectively. Then the velocity profiles (u) for different zones may be obtained from the empirical relations using dimensional analysis. 
Wall layer : In this region, it is approximated that u is independent of shear layer thickness so that the following empirical relation holds good. 
Eq. (5.1.12) is known as the law of wall and the quantity is called as friction velocity. It should not be confused with flow velocity. 
Outer layer : The velocity profile in the outer layer is approximated as the deviation from the free stream velocity and represented by an equation called as velocity-defect law 
Overlap layer : Most of the experimental data show the very good validation of wall law and velocity defect law in the respective regions. An intermediate layer may be obtained when the velocity profiles described by Eqs. (5.1.12 & 5.1.13) overlap smoothly. It is shown that empirically that the overlap layer varies logarithmically with y (Eq. (5.1.14). This particular layer is known as overlap layer