Universal Velocity Distribution Law & Friction Factor in Duct Flows - 2 | Fluid Mechanics for Mechanical Engineering PDF Download

Fully Developed Turbulent Flow In A Pipe For Moderate Reynolds Numbers 

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

From equation (34.23)   

                                                                        (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.

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

                                 (34.24b)

where u_r is friction velocity. However,  may be spitted into     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,f, defined by Eq. (34.22) is valid for a specific range of Reynolds number and for a particular surface condition.
   
    

   Skin Friction Coefficient For Boundary Layers On A Flat Plate

• Calculations of skin friction drag on lifting surface and on aerodynamic bodies are somewhat similar to the analyses of skin friction on a flat plate. Because of zero pressure gradient, the flat plate at zero incidence is easy to consider. In some of the applications cited above, the pressure gradient will differ from zero but the skin friction will not be dramatically different so long there is no separation. 

• We begin with the momentum integral equation for flat plate boundary layer which is valid for both laminar and turbulent flow.

                                           (34.26a)

nvoking the definition of   Eq.(34.26a) can be written as

                                                (34.26b)

• Due to the similarity in the laws of wall, correlations of previous section may be applied to the flat plate by substituting δ for R and U_∞ for the time mean velocity at the pipe centre.The rationale for using the turbulent pipe flow results in the situation of a turbulent flow over a flat plate is to consider that the time mean velocity, at the centre of the pipe is analogous to the free stream velocity, both the velocities being defined at the edge of boundary layer thickness.

             Finally, the velocity profile will be [following Eq. (34.24)]

                                     (34.27)              

Evaluating momentum thickness with this profile, we shall obtain

                         (34.28)

Consequently, the law of shear stress (in range o  for the flat plate is found out by making use of the pipe flow expression of Eq. (34.24b) as

Substituting U_∞ for  and δ  for R in the above expression, we get

                                            (34.29)

Once again substituting Eqs (34.28) and (34.29) in Eq.(34.26), we obtain

                                  (34.30)         

Continued...Skin Friction Coefficient For Boundary Layers On A Flat Plate

• For simplicity, if we assume that the turbulent boundary layer grows from the leading edge of the plate we shall be able to apply the boundary conditions x = 0, δ = 0 which will yield C = 0, and Eq. (34.30) will become From Eqs (34.26b), (34.28) and (34.31), it is possible to calculate the average skin friction coefficient on a flat plate as

                                    (34.31)

Where                                          

From Eqs (34.26b), (34.28) and (34.31), it is possible to calculate the average skin friction coefficient on a flat plate as

                                            (34.32)

                     It can be shown that Eq. (34.32) predicts the average skin friction coefficient correctly in the regime of Reynolds number below 

2 x 10⁶.

• This result is found to be in good agreement with the experimental results in the range of Reynolds number between 5 x 10⁵and 10⁷ which is given by

                                          (34.33)

        Equation (34.33) is a widely accepted correlation for the average value of turbulent skin friction coefficient on a flat plate.

• With the help of Nikuradse's experiments, Schlichting obtained the semi empirical equation for the average skin friction coefficient as

                                               (34.34)  

Equation (34.34) was derived asssuming the flat plate to be completely turbulent over its entire length . In reality, a portion of it is laminar from the leading edge to some downstream position. For this purpose, it was suggested to use

                               (34.35a)   

          where A has various values depending on the value of Reynolds number at which the transition takes place.

• If the trasition is assumed to take place around a Reynolds number of 5 x 10⁵, the average skin friction correlation of Schlichling can be written as

                           (34.35b)

          All that we have presented so far, are valid for a smooth plate.

• Schlichting used a logarithmic expression for turbulent flow over a rough surface and derived

                    (34.36)