Compressibility transformations have received considerable attention for extending well-established incompressible wall models to high-speed flows. While encouraging progress has been made in mean velocity scalings, research on temperature transformations has lagged behind. In this study, we rigorously derive a general framework for both velocity and temperature transformations directly from the compressible Reynolds-averaged Navier–Stokes (RANS) equations and their ‘incompressible’ counterparts, elucidating how these transformations guide the development of compressible algebraic RANS models in the inner layer. The introduction of the mixed Prandtl number further links the mean momentum and energy transport, facilitating the formulation of novel temperature transformations through integration with arbitrary mean velocity scalings, thereby unifying existing transformation methods while providing a systematic approach for further improvement. A detailed evaluation using direct numerical simulation databases of canonical compressible wall-bounded turbulent flows (CWBTFs) demonstrates that temperature transformations based on the Griffin–Fu–Moin and our recently proposed velocity scalings exhibit superior accuracy and robustness across a wide range of Reynolds and Mach numbers, as well as varying wall thermal boundary conditions. We also perform a preliminary investigation into the applicability of the proposed integral mean temperature–velocity relation and inverse temperature transformations for near-wall temperature modelling in cold-wall boundary layer flows, where discontinuities caused by non-monotonic temperature distributions are effectively avoided. Although the omission of higher-order terms in deriving the total heat flux equation enables closed-form wall modelling, it remains a key limitation to the model’s accuracy at the current stage. Future work may therefore need to address this issue to achieve further advances. These findings enhance the physical understanding of mean momentum and energy transport in canonical CWBTFs, and offer promising prospects for advancing near-wall temperature modelling within RANS and wall-modelled large eddy simulation frameworks.
Zhu et al. (Wed,) studied this question.