The average velocity for this profile can be computed as,

(5.5.3)

Using the approximate values of , the simplified relation for turbulent velocity profile is obtained as below;

(5.5.4)

Recall the Darcy friction factor which relates the wall shear stress and average velocity ;

(5.5.5)

Rearrange the first term appearing in RHS of Eq. (5.5.4)

(5.5.6)

Substituting Eqs. (5.5.5 & 5.5.6) in Eq. (5.5.4) and simplifying, one can get the following relation for friction factor for the turbulent pipe flow.

(5.5.7)

Since, Eq. (5.5.7) is implicit in nature, it becomes cumbersome to obtain friction factor for a given Reynolds number. So, there are many alternative explicit approximations as given below;

(5.5.8)

Further, the maximum velocity in the turbulent pipe flow is obtained from (5.5.2) and is evaluated at ;

(5.5.9)

Another correlation may be obtained by relating Eq. (5.5.9) with the average velocity (Eq. 5.5.3);

(5.5.10)