Velocity and temperature distributions are both crucial for modeling compressible wall-bounded turbulent flows. The compressible law of the wall for velocity has been extensively examined through velocity transformations. However, a well-established temperature transformation remains an open issue. We propose new Van Driest type (VD-type) and semi-local type (SL-type) temperature transformation for compressible turbulent channel flow. Our approach is based on an analysis of the momentum and energy balance equations in the overlap layer. It accounts for the influences of mixing length model, the work of the body force, and the turbulent kinetic energy (TKE) flux. The proposed transformations are evaluated using data from direct numerical simulations and wall-resolved large eddy simulations of compressible turbulent channel flow. The SL-type transformation provides better data collapse than the VD-type in the viscous sublayer and buffer layer. With a suitable mixing length model, the SL-type transformed temperature agrees well with the incompressible temperature profile or the extended law of the wall. For the isothermal wall, the integral mean error over the entire boundary layer remains below 2% for most cases, with root mean square value of about 1.7%. The results highlight the importance of mitigating the energy imbalance in the transformation. This work identifies the multi-layer structure of the turbulent TKE flux, which in turn enables approximate models and corresponding simplified yet effective temperature transformations. Applications of the proposed approach in near-wall modeling and inverse transformation, as well as its potential extension to more general configurations, are also discussed.
Xu et al. (Sun,) studied this question.