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 their assessment has largely focused on the apparent collapse of transformed mean velocity profiles, without enforcing a unique, Mach-independent representation of the mean shear. In this work, a stricter consistency condition is proposed, requiring that a single incompressible inner--outer model for the mean velocity gradient reproduce all transformed compressible profiles when expressed in terms of a transformed wall-normal coordinate. This condition implies collapse not only of the transformed mean velocity but also of semilocal eddy viscosity and turbulent kinetic energy production. Existing compressibility transformations are shown, using hypersonic direct numerical simulation data, to incur velocity errors of $1$--$25\%$ relative to the chosen incompressible inner--outer model, particularly for strongly cooled cases. A new forward compressible-to-incompressible transformation is then developed that constructs the transformed coordinate as a convex combination of semilocal and integral-type basis functions with coefficients modeled as functions of friction Mach number $M_\tau$ and wall heat transfer rate $B_q$. Casewise optimization yields consistency errors of $1$--$4\%$ across the available hypersonic direct numerical simulation database, and this performance is retained using simple multi-linear and multi-quadratic regressions in $(M_\tau,B_q)$. The forward transformation is subsequently embedded in an inverse incompressible-to-compressible transformation framework, which reconstructs the mean compressible state from freestream and wall conditions at a prescribed boundary layer thickness. The inverse solver recovers several key boundary layer parameters, velocity profiles, and skin friction distributions with good accuracy, and generally improves upon existing models for cold-wall hypersonic TBLs, thereby providing a physically constrained basis for near-wall modeling in hypersonic turbulent boundary layers with strong wall cooling.