Predictive Compressibility Transformation for Hypersonic Turbulent Boundary Layers with Cold Walls

Compressibility transformations are widely used to relate hypersonic zero-pressure-gradient turbulent boundary layers to incompressible reference states, but they are usually assessed only by the collapse of transformed mean velocity profiles alone. Here, a stricter consistency requirement is proposed: the transformed velocity profile is required to agree pointwise with the profile obtained from a fixed incompressible inner–outer model at the same transformed wall-normal coordinate. A second requirement is that the same coordinate should also improve the collapse of semilocal eddy viscosity and turbulent kinetic energy production profiles without modifying their traditional definitions. Using hypersonic direct numerical simulation data under these criteria, existing transformations are shown to incur velocity errors of 1–28%, especially for strongly cooled walls. A new forward transformation is formulated by correcting the semilocal coordinate and adding a corrected wake contribution to the transformed velocity. The proposed transformation satisfies the proposed consistency requirements for a wide range of hypersonic zero-pressure-gradient turbulent boundary-layer cases with 1.2–4.5% velocity errors with minimal calibration. Embedded in an inverse incompressible-to-compressible framework driven by freestream and wall conditions at a prescribed boundary-layer thickness, this model reconstructs key boundary-layer parameters, skin friction, and mean velocity profiles with 2.2–7.5% velocity errors, demonstrating its predictive capability for cold-wall hypersonic turbulent boundary layers.