# 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 [1]):

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

Schlichting (equation 21.16 footnote in [1])

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

Schultz-Grunov (equation 21.19a in [1]):

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

(equation 38 in [2]):

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

## References

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

## To do

Someone should add more data about total skin friction approximations, Prandtl-Schlichting skin-friction formula, and the Karman-Schoenherr equation.