In the present study, we report a new Mach number invariant function ($\phi_{S}$) for the mean temperature field in compressible wall turbulence. We demonstrate its validation by comparing it with the invariant functions derived in the previous studies, i.e., the semi-local-type ($\phi_{SL}$) and van-Driest-type ($\phi_{VD}$) scalings, case by case. To be specific, $\phi_{SL}$ works well in the inner layer of compressible channel flows with isothermal walls; $\phi_{S}$ works well in the inner layer of compressible channel flows with adiabatic walls, and supersonic/hypersonic turbulent boundary layers with cold walls; $\phi_{VD}$ does not work the best among all three functions in the flows under consideration. The newly proposed temperature transformations based on $\phi_{S}$ show an improvement in channel flows over adiabatic walls and supersonic/hypersonic turbulent boundary layers with cold walls. The effects of the generated high-order terms during derivation are also clarified. These findings may be revealing for the development of the near-wall model in high-speed aerodynamics.
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