van Driest II Method
The van Driest II method maps a compressible turbulent boundary layer to an equivalent incompressible one, evaluates the incompressible turbulent skin friction law, then maps back to compressible form1.
The turbulent mapping structure is documented in Turbulent Estimates.
The van Driest II correction factors are
\[
F_c = \frac{rm}{\left[\sin^{-1}(\alpha) + \sin^{-1}(\beta)\right]^2},
\]
\[
F_x = \frac{\mu_e/\mu_w}{F_c}.
\]
where \(\mu_e = \mu(T_e)\) and \(\mu_w = \mu(T_w)\).
\(\alpha\) and \(\beta\) are defined as
\[
\alpha = \frac{2a^2 - b}{D},
\quad
\beta = \frac{b}{D}.
\]
\(a\), \(b\), and \(D\) are
\[
a = \sqrt{\frac{mr}{F}},
\quad
b = \frac{1 + rm - F}{F},
\quad
D = \sqrt{4a^2 + b^2},
\]
with \(m\), \(r\), and \(F\) as
\[
m = \frac{\gamma-1}{2}M_e^2,
\quad
r = \mathrm{Pr}^{1/3},
\quad
F = \frac{T_w}{T_e}.
\]
References
-
Van Driest, E.R. (1951). Turbulent boundary layer in compressible fluids. Journal of the Aeronautical Sciences, 18(3), 145-160. ↩