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Turbulent Estimates

The turbulent skin-friction estimates map the incompressible Karman-Schoenherr relation1 to the compressible case using the correction factors \(F_x\) and \(F_c\). The factor \(F_x\) maps the Reynolds number to an equivalent incompressible Reynolds number.

The incompressible turbulent average skin-friction relation by Karman-Schoenherr 1 is:

\[ \frac{0.242}{\sqrt{C_{f,\mathrm{inc}}}} = \log_{10}\!\left(\mathrm{Re}_{x,\mathrm{inc}}\,C_{f,\mathrm{inc}}\right). \]

The average coefficient is converted to the local incompressible coefficient with the Hopkins-Inouye relation2:

\[ c_{f,\mathrm{inc}} = \frac{0.242\,C_{f,\mathrm{inc}}} {0.242 + 0.8686\sqrt{C_{f,\mathrm{inc}}}}. \]

The mapping is done with the correction factors \(F_x\) and \(F_c\):

\[ \mathrm{Re}_{x,\mathrm{inc}} = F_x\,\mathrm{Re}_x \]
\[ c_f = \frac{c_{f,\mathrm{inc}}}{F_c}. \]

For all turbulent methods, the Stanton number is then computed from Reynolds analogy:

\[ C_h = \frac{c_f}{2\,\mathrm{Pr}^{2/3}}. \]

The only method-specific part is how \(F_x\) and \(F_c\) are computed:

References

  1. Schoenherr, K.E. (1932). Resistance of flat surfaces moving through a fluid. Trans. SNAME, 40, 279-313. ↩↩

  2. Hopkins, E.J. and Inouye, M. (1971). An evaluation of theories for predicting turbulent skin friction and heat transfer on flat plates at supersonic and hypersonic Mach numbers. NASA TN D-6353. ↩