# Skin friction coefficient

The skin friction coefficient, $C_f$, is defined by:

$C_f \equiv \frac{\tau_w}{\frac{1}{2} \, \rho \, U_\infty^2}$

Where $\tau_w$ is the local wall shear stress, $\rho$ is the fluid density and $U_\infty$ is the free-stream velocity (usually taken ouside of the boundary layer or at the inlet).

For a turbulent boundary layer several approximation formulas for the local skin friction for a flat plate can be used:

1/7 power law:

$C_f = 0.0576 Re_x^{-1/5} \quad \mbox{for} \quad 5 \cdot 10^5 < Re_x < 10^7$

1/7 power law with experimental calibration (equation 21.12 in [3]):

$C_f = 0.0592 \, Re_x^{-1/5} \quad \mbox{for} \quad 5 \cdot 10^5 < Re_x < 10^7$

Schlichting (equation 21.16 footnote in [3])

$C_f = [ 2 \, log_{10}(Re_x) - 0.65 ] ^{-2.3} \quad \mbox{for} \quad Re_x < 10^9$

Schultz-Grunov (equation 21.19a in [3]):

$C_f = 0.370 \, [ log_{10}(Re_x) ]^{-2.584}$

(equation 38 in [1]):

$1.0/C_f^{1/2} = 1.7 + 4.15 \, log_{10} (Re_x \, C_f)$

The following skin friction formulas are extracted from [2],p.19. Proper reference needed:

Prandtl (1927):

$C_f = 0.074 \, Re_x^{-1/5}$

Telfer (1927):

$C_f = 0.34 \, Re_x^{-1/3} + 0.0012$

Prandtl-Schlichting (1932):

$C_f = 0.455 \, [ log_{10}(Re_x)]^{-2.58}$

Schoenherr (1932):

$C_f = 0.0586 \, [ log_{10}(Re_x \, C_f )]^{-2}$

Schultz-Grunov (1940):

$C_f = 0.427 \, [ log_{10}(Re_x) - 0.407]^{-2.64}$

Kempf-Karman (1951):

$C_f = 0.055 \, Re_x^{-0.182}$

Lap-Troost (1952):

$C_f = 0.0648 \, [log_{10}(Re_x \, C_f^{0.5})-0.9526]^{-2}$

Landweber (1953):

$C_f = 0.0816 \, [log_{10}(Re_x) - 1.703]^{-2}$

Hughes (1954):

$C_f = 0.067 \, [log_{10}(Re_x) - 2 ] ^{-2}$

Wieghard (1955):

$C_f = 0.52 \, [log_{10}(Re_x)] ^{-2.685}$

ITTC (1957):

$C_f = 0.075 \, [log_{10}(Re_x) - 2 ] ^{-2}$

$C_f = 0.0113 \, [log_{10}(Re_x) - 3.7 ] ^{-1.15}$

Granville (1977):

$C_f = 0.0776 \, [log_{10}(Re_x) - 1.88 ] ^{-2} + 60 \, Re_x^{-1}$

Date Turnock (1999):

$C_f = [4.06 \, log_{10}(Re_x \, C_f) - 0.729]^{-2}$

## References

1. von Karman, Theodore (1934), "Turbulence and Skin Friction", J. of the Aeronautical Sciences, Vol. 1, No 1, 1934, pp. 1-20.
3. Schlichting, Hermann (1979), Boundary Layer Theory, ISBN 0-07-055334-3, 7th Edition.

## To do

Someone should add more data about total skin friction approximations, Prandtl-Schlichting skin-friction formula, and the Karman-Schoenherr equation. Add proper reference for equations in [2]

Edit: With regards to the 1/7th power law, in Schlichtings book (see references) the formula describing Cf over a flat plate , without pressure gradient, is Cf=0.0725*Re^(-1/5) and it is valid between 5x10^5<Re<10^7 with the assumption of the flow being turbulent from the leading edge (page 639) This is found in page 638 , formula 21.11.

Taking into account that the flow is laminar for the first part of the plate and using Blasius's equeation, after providing corrective some corrective factors , Schlichting in page 644 states: Cf=0.02666*Rl^(-0.139) There should be a separation between local and total skin friction on the plate.grizos