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

Schlichting (equation 21.16 footnote in ) $C_f = [ 2 \, log_{10}(Re_x) - 0.65 ] ^{-2.3} \quad \mbox{for} \quad Re_x < 10^9$

Schultz-Grunov (equation 21.19a in ): $C_f = 0.370 \, [ log_{10}(Re_x) ]^{-2.584}$

(equation 38 in ): $1.0/C_f^{1/2} = 1.7 + 4.15 \, log_{10} (Re_x \, C_f)$

The following skin friction formulas are extracted from ,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): $C_f = [4.06 \, log_{10}(Re_x \, C_f) - 0.729]^{-2}$