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Eckert Reference-Temperature Method

This method is a compressible correction of the Blasius laminar estimate, applied through a reference temperature \(T^*\).

It keeps the same Blasius skin-friction form and replaces \(\mathrm{Re}_x\) with a corrected local Reynolds number \(\mathrm{Re}_x^*\).

The reference temperature based on 12 is

\[ \frac{T^*}{T_e} = 0.5 + 0.039 M_e^2 + 0.5\frac{T_w}{T_e} \]
Alternative reference temperature equations

A common alternative form of the reference temperature written with recovery temperature is

\[ T^* = T_e + 0.5\,(T_w - T_e) + 0.22\,(T_r - T_e) \]

with

\[ T_r - T_e = r\,\frac{\gamma-1}{2}\,M_e^2\,T_e \]

so

\[ \frac{T^*}{T_e} = 0.5 + 0.5\frac{T_w}{T_e} + 0.22\,r\,\frac{\gamma-1}{2}\,M_e^2 \]

For air (for example \(\gamma=1.4\) and \(r\approx \mathrm{Pr}^{1/3}\approx0.89\)), this simplifies to

\[ 0.22\,r\,\frac{\gamma-1}{2} \approx 0.22\times0.89\times0.2 \approx 0.039 \]

The local Reynolds number is then corrected as follows:

\[ \mathrm{Re}_x^* = \mathrm{Re}_x\left(\frac{\rho^*}{\rho_e}\right)\left(\frac{\mu_e}{\mu^*}\right) \]

and the skin-friction estimate based on the Eckert reference temperature method is:

\[ c_f = \frac{0.664}{\sqrt{\mathrm{Re}_x^*}} \]

The Stanton number is computed with the Reynolds analogy:

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

References

  1. Eckert, E.R.G. (1955). Engineering relations for heat transfer and friction in high-velocity laminar and turbulent boundary-layer flow over surfaces with constant pressure and temperature. Trans. ASME, 78, 1273-1283. ↩

  2. Eckert, E.R.G. (1955). Engineering relations for friction and heat transfer to surfaces in high velocity flow. ↩