Fully Developed Turbulent Flow In A Pipe For Moderate Reynolds Numbers

• 
  The ratio of  and  for the aforesaid profile is found out by considering the volume flow rate Q as

      

From equation (34.23)  

 

or

or

or

or

(34.24a)

• Now, for different values of n (for different Reynolds numbers) we shall obtain different values of  from Eq.(34.24a). On substitution of Blasius resistance formula (34.22) in Eq.(34.21), the following expression for the shear stress at the wall can be obtained.

putting                      

and where       

or

or

• For n=7,  becomes equal to 0.8. substituting  in the above equation, we get

Finally it produces

(34.24b)

or

where  is friction velocity. However,  may be spitted into  and  and we obtain

or

(34.25a)

• Now we can assume that the above equation is not only valid at the pipe axis (y = R) but also at any distance from the wall y and a general form is proposed as

(34.25b)

• Concluding Remarks :

1. It can be said that (1/7)th power velocity distribution law (24.38b) can be derived from Blasius's resistance formula (34.22) . 
2. Equation (34.24b) gives the shear stress relationship in pipe flow at a moderate Reynolds number, i.e  . Unlike very high Reynolds number flow, here laminar effect cannot be neglected and the laminar sub layer brings about remarkable influence on the outer zones. 
3. The friction factor for pipe flows,, defined by Eq. (34.22) is valid for a specific range of Reynolds number and for a particular surface condition.