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Skin friction coefficient

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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).
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).
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For a turbulent boundary layer several approximation formulas for the local skin friction for a flat plate can be used:
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It is related to the momentum thickness as follows: C_f = 2(d theta)/ (d x)
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1/7 power law:
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An empirical relation you may use for comparison is: C_f = 0.0583/(Re )^0.2
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:<math>C_f = 0.0576 Re_x^{-1/5} \quad \mbox{for} \quad 5 \cdot 10^5 < Re_x < 10^7 </math>
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1/7 power law with experimental calibration (equation 21.12 in [[#References|[3]]]):
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''Someone should add some correlations and references for them here''
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:<math>C_f = 0.0592 \, Re_x^{-1/5} \quad \mbox{for} \quad 5 \cdot 10^5 < Re_x < 10^7</math>
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Schlichting (equation 21.16 footnote in [[#References|[3]]])
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:<math>C_f = [ 2 \, log_{10}(Re_x) - 0.65 ] ^{-2.3} \quad \mbox{for} \quad Re_x < 10^9 </math>
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Schultz-Grunov (equation 21.19a in [[#References|[3]]]):
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:<math>C_f = 0.370 \, [ log_{10}(Re_x) ]^{-2.584} </math>
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(equation 38 in [[#References|[1]]]):
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:<math>1.0/C_f^{1/2} = 1.7 + 4.15 \, log_{10} (Re_x \, C_f) </math>
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The following skin friction formulas are extracted from [[#References|[2]]],p.19. Proper reference needed:
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Prandtl (1927):
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:<math> C_f = 0.074 \, Re_x^{-1/5} </math>
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Telfer (1927):
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:<math> C_f = 0.34 \, Re_x^{-1/3} + 0.0012 </math>
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Prandtl-Schlichting (1932):
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:<math> C_f = 0.455 \, [ log_{10}(Re_x)]^{-2.58} </math>
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Schoenherr (1932):
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:<math> C_f = 0.0586 \, [ log_{10}(Re_x \, C_f )]^{-2} </math>
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Schultz-Grunov (1940):
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:<math> C_f = 0.427 \, [ log_{10}(Re_x) - 0.407]^{-2.64} </math>
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Kempf-Karman (1951):
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:<math> C_f = 0.055 \, Re_x^{-0.182} </math>
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Lap-Troost (1952):
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:<math> C_f = 0.0648 \, [log_{10}(Re_x \, C_f^{0.5})-0.9526]^{-2} </math>
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Landweber (1953):
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:<math> C_f = 0.0816 \, [log_{10}(Re_x) - 1.703]^{-2} </math>
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Hughes (1954):
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:<math> C_f = 0.067 \, [log_{10}(Re_x) - 2 ] ^{-2} </math>
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Wieghard (1955):
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:<math> C_f = 0.52 \, [log_{10}(Re_x)] ^{-2.685} </math>
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ITTC (1957):
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:<math> C_f = 0.075 \, [log_{10}(Re_x) - 2 ] ^{-2} </math>
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Gadd (1967):
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:<math> C_f = 0.0113 \, [log_{10}(Re_x) - 3.7 ] ^{-1.15} </math>
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Granville (1977):
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:<math> C_f = 0.0776 \, [log_{10}(Re_x) - 1.88 ] ^{-2} + 60 \, Re_x^{-1} </math>
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Date Turnock (1999):
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:<math> C_f = [4.06 \, log_{10}(Re_x \, C_f) - 0.729]^{-2} </math>
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== References ==
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# {{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}}
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# {{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])}}
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# {{reference-book|author=Schlichting, Hermann |year=1979|title=Boundary Layer Theory|rest=ISBN 0-07-055334-3, 7th Edition}}
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== To do ==
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''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}}
{{stub}}

Latest revision as of 06:52, 23 September 2011

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

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


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