Skin friction coefficient

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Latest revision as of 12:14, 14 January 2016

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):

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

Gadd (1967):

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

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

von Karman, Theodore (1934), "Turbulence and Skin Friction", J. of the Aeronautical Sciences, Vol. 1, No 1, 1934, pp. 1-20.
Lazauskas, Leo Victor (2005), "Hydrodynamics of Advanced High-Speed Sealift Vessels", Master Thesis, University of Adelaide, Australia (download).
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]

This article is a stub, a short article which needs to be improved. You can help by expanding it.

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

@@ Line 5: / Line 5: @@
 Where <math>\tau_w</math> is the local [[wall shear stress]], <math>\rho</math> is the fluid density and <math>U_\infty</math> 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 can be used:
+For a turbulent boundary layer several approximation formulas for the local skin friction for a flat plate can be used:
 /7 power law:
-<math>C_f = 0.0576 Re_x^{-1/5} \quad \mbox{for} \quad 5 \cdot 10^5 < Re_x < 10^7 </math>
+:<math>C_f = 0.0576 Re_x^{-1/5} \quad \mbox{for} \quad 5 \cdot 10^5 < Re_x < 10^7 </math>
-/7 power law with experimental calibration (equation 21.12 in [1]):
+/7 power law with experimental calibration (equation 21.12 in [[#References|[3]]]):
-<math>C_f = 0.0592 \, Re_x^{-1/5} \quad \mbox{for} \quad 5 \cdot 10^5 < Re_x < 10^7</math>
+:<math>C_f = 0.0592 \, Re_x^{-1/5} \quad \mbox{for} \quad 5 \cdot 10^5 < Re_x < 10^7</math>
-Schlichting (equation 21.16 footnote in [1])
+Schlichting (equation 21.16 footnote in [[#References|[3]]])
-<math>C_f = [ 2 \, log(Re_x) - 0.65 ] ^{-2.3} \quad \mbox{for} \quad Re_x < 10^9 </math>
+:<math>C_f = [ 2 \, log_{10}(Re_x) - 0.65 ] ^{-2.3} \quad \mbox{for} \quad Re_x < 10^9 </math>
-Schultz-Grunov (equation 21.19a in [1]):
+Schultz-Grunov (equation 21.19a in [[#References|[3]]]):
-<math>C_f = 0.370 \, [ log(Re_x) ]^{-2.584} </math>
+:<math>C_f = 0.370 \, [ log_{10}(Re_x) ]^{-2.584} </math>
-== References ==
+(equation 38 in [[#References|[1]]]):
+:<math>1.0/C_f^{1/2} = 1.7 + 4.15 \, log_{10} (Re_x \, C_f) </math>
+The following skin friction formulas are extracted from [[#References|[2]]],p.19. Proper reference needed:
+Prandtl (1927):
+:<math> C_f = 0.074 \, Re_x^{-1/5} </math>
+Telfer (1927):
+:<math> C_f = 0.34 \, Re_x^{-1/3} + 0.0012 </math>
+Prandtl-Schlichting (1932):
+:<math> C_f = 0.455 \, [ log_{10}(Re_x)]^{-2.58} </math>
+Schoenherr (1932):
+:<math> C_f = 0.0586 \, [ log_{10}(Re_x \, C_f )]^{-2} </math>
+Schultz-Grunov (1940):
+:<math> C_f = 0.427 \, [ log_{10}(Re_x) - 0.407]^{-2.64} </math>
+Kempf-Karman (1951):
+:<math> C_f = 0.055 \, Re_x^{-0.182} </math>
+Lap-Troost (1952):
+:<math> C_f = 0.0648 \, [log_{10}(Re_x \, C_f^{0.5})-0.9526]^{-2} </math>
+Landweber (1953):
+:<math> C_f = 0.0816 \, [log_{10}(Re_x) - 1.703]^{-2} </math>
+Hughes (1954):
+:<math> C_f = 0.067 \, [log_{10}(Re_x) - 2 ] ^{-2} </math>
+Wieghard (1955):
+:<math> C_f = 0.52 \, [log_{10}(Re_x)] ^{-2.685} </math>
+ITTC (1957):
+:<math> C_f = 0.075 \, [log_{10}(Re_x) - 2 ] ^{-2} </math>
+Gadd (1967):
+:<math> C_f = 0.0113 \, [log_{10}(Re_x) - 3.7 ] ^{-1.15} </math>
+Granville (1977):
+:<math> C_f = 0.0776 \, [log_{10}(Re_x) - 1.88 ] ^{-2} + 60 \, Re_x^{-1} </math>
+Date Turnock (1999):
+:<math> C_f = [4.06 \, log_{10}(Re_x \, C_f) - 0.729]^{-2} </math>
+== References ==
+# {{reference-paper|author=von Karman, Theodore |year=1934|title=Turbulence and Skin Friction|rest=J. of the Aeronautical Sciences, Vol. 1, No 1, 1934, pp. 1-20}}
+# {{reference-paper|author=Lazauskas, Leo Victor |year=2005|title=Hydrodynamics of Advanced High-Speed Sealift Vessels|rest=Master Thesis, University of Adelaide, Australia ([http://digital.library.adelaide.edu.au/dspace/bitstream/2440/37729/1/02whole.pdf download])}}
 # {{reference-book|author=Schlichting, Hermann |year=1979|title=Boundary Layer Theory|rest=ISBN 0-07-055334-3, 7th Edition}}
@@ Line 30: / Line 80: @@
 ''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]''
 {{stub}}
+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

Skin friction coefficient

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Latest revision as of 12:14, 14 January 2016

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