Accurate scaling of heat energy is critical for compressible turbulence modeling. In addition to the temperature-velocity relation, temperature transformations have received increasing interest recently, but their overall accuracy is still inferior to the velocity counterparts. In this work, we propose to use total enthalpy $H$ instead of temperature to construct the compressible law of the wall for heat energy. A critical justification is the structural resemblance between the equations of compressible $H$ and incompressible passive scalars. Also, $H$ reduces to static enthalpy at zero Mach number, meeting the requirement of self-consistency for transformations. To build the transformation, we first define and uncover the scalings of laminar and turbulent Prandtl numbers for $H$, and then reveal the strong analogy of heat flux budgets between compressible $H$ and incompressible passive scalars. Finally, a semilocal total enthalpy transformation of no fitted parameters is constructed for channel flows, which is structurally analogous and comparable in accuracy to the Trettel-Larsson velocity transformation. This new transformation can potentially serve as a wall model for energy equations to accurately predict the enthalpy or temperature in compressible turbulence.

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• Boundary layers
• Compressible boundary layers
• Turbulent boundary layers