Paper reference: Zhang, Y.-S.; Bi, W.-T.; Hussain, F. & She, Z.-S. A generalized Reynolds analogy for compressible wall-bounded turbulent flows, J. Fluid Mech., Cambridge University Press (CUP), 2013 , 739 , 392-420.
The mean temperature is a quadratic function of the mean velocity.
\[ \begin{aligned} \frac{T}{T_{\infty}} &= \frac{T_{w}}{T_{\infty}} + \frac{T_{aw}-T_{w}}{T_{\infty}} \frac{U}{U_{\infty}} + \frac{T_{\infty}-T_{aw}}{T_{\infty}} \left ( \frac{U}{U_{\infty}} \right )^2 \\ T_{aw} &= T_{\infty} + r \frac{U^2_{\infty}}{2 C_p} = 1 + r \frac{\gamma-1}{2} M^2_{\infty} \\ \end{aligned} \]
In the original Crocco-Busemann relation, the recovery factor is 1. The modified version sets \(r=0.89\). The modifed version is also called Walz’s equation.
The modified Crocco-Busemann relation agrees with DNS very well in adiabatic compressible turbulent boundary layers. In the disbatic wall conditions, it deviates from DNS clearly.
Replace \(U/U_{\infty}\) with \(f \left ( U/U_{\infty} \right )\)
\[ \begin{aligned} \frac{T}{T_{\infty}} &= \frac{T_{w}}{T_{\infty}} + \frac{T_{aw}-T_{w}}{T_{\infty}} f \left ( \frac{U}{U_{\infty}} \right ) + \frac{T_{\infty}-T_{aw}}{T_{\infty}} \left ( \frac{U}{U_{\infty}} \right )^2 \\ f \left ( \frac{U}{U_{\infty}} \right ) &= (1-\alpha) \left ( \frac{U}{U_{\infty}} \right )^2 + \alpha \left ( \frac{U}{U_{\infty}} \right ), \alpha=0.8259 \\ \end{aligned} \]
\[ \begin{aligned} \frac{T}{T_{\infty}} &= \frac{T_{w}}{T_{\infty}} + \frac{T_{aw}-T_{w}}{T_{\infty}} f \left ( \frac{U}{U_{\infty}} \right ) + \frac{T_{\infty}-T_{aw}}{T_{\infty}} \left ( \frac{U}{U_{\infty}} \right )^2 \\ f \left ( \frac{U}{U_{\infty}} \right ) &= (1-\alpha) \left ( \frac{U}{U_{\infty}} \right )^2 + \alpha \left ( \frac{U}{U_{\infty}} \right ) \\ \alpha &= s Pr \\ \end{aligned} \]
where \(s\) is related to the wall heat transfer.