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)

  Invoking 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  for R and  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)]

for

(34.27)

Evaluating momentum thickness with this profile, we shall obtain

(34.28)

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

Substituting  for  and  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

	.
	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 yieldC = 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   or,      or,    (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 . This result is found to be in good agreement with the experimental results in the range of Reynolds number between  and  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 , 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)