A general Reynolds analogy theory for the compressible wall-bounded   turbulence
  A general Reynolds analogy (GRA) theory is proposed for the mean and fluctuating velocity and temperature in compressible wall-bounded turbulent flows. In particular, an exact analogy solution is derived for compressible turbulent pipe and channel flows and an approximate analogy solution is derived for compressible turbulent boundary layers (CTBL), both of which are independent of fluid Prandtl number and wall temperature condition. The analogy solutions are in excellent agreement with direct numerical simulation data, able to reproduce empirical relations, and can be viewed as extensions of existing theories. In contrast to Walz's equation for adiabatic CTBL, the mean temperature-velocity relation derived by GRA can be applied to different wall-bounded flows in non-adiabatic wall condition, which is achieved by extending Walz's adiabatic recovery factor to a heat flux dependent one. The fluctuation temperature-velocity relations derived by GRA are slightly different from the modified strong Reynolds analogy derived phenomenologically by Huang et al. (HSRA), and have a better performance than HSRA. In addition, several key quantities are introduced in GRA, including a general total enthalpy (or temperature) and an adiabatic degree--a well-founded dimensionless parameter for characterizing the wall-temperature effects in non-adiabatic flows. The GRA unveils the universal feature behind the complex nonlinear couplings between the thermal and velocity fields, and makes possible of predicting the mean fields of compressible wall-bounded turbulence with the information of the corresponding incompressible flow. 
You-sheng Zhang, Wei-tao Bi, Fazle. Husain, Xin-liang Li, and Zhen-su   She
physics.flu-dyn