White-Christoph Method
The White-Christoph method is a simplified closed-form approximation to van Driest II-style turbulent compressibility transformation1.
The shared turbulent mapping structure is documented in Turbulent Estimates.
The White-Christoph correction factors are
\[
F_c = S^2,
\]
\[
F_x = \frac{\mu_e}{\mu_w}\,\frac{1}{\sqrt{F}\,S},
\]
where \(\mu_e = \mu(T_e)\) and \(\mu_w = \mu(T_w)\).
The required factors are defined as
\[
m = \frac{\gamma-1}{2}M_e^2,
\qquad
r = \mathrm{Pr}^{1/3},
\]
and
\[
F = \frac{T_w}{T_e},
\qquad
\frac{T_{aw}}{T_e} = 1 + rm,
\qquad
F_{aw} = \frac{T_w}{T_{aw}} = \frac{F}{T_{aw}/T_e}.
\]
With those the following values can be computed
\[
a = \sqrt{\frac{m}{F}},
\qquad
b = \frac{1 - F_{aw}}{F_{aw}},
\qquad
D = \sqrt{4a^2 + b^2},
\]
\[
\alpha = \frac{2a^2 - b}{D},
\qquad
\beta = \frac{b}{D},
\]
\[
S = \frac{\sqrt{F/F_{aw} - 1}}{\sin^{-1}(\alpha) + \sin^{-1}(\beta)}.
\]
References
-
White, F.M. and Christoph, G.H. (1972). A simple new analysis of the turbulent compressible boundary layer. AIAA Paper 70-164. ↩