Predictive Compressibility Transformation for Hypersonic Turbulent Boundary Layers with Cold Walls

Abstract: Compressibility transformations are used to relate hypersonic zero-pressure-gradient (ZPG) turbulent boundary layers (TBLs) to incompressible reference states, but their assessment has largely focused on the 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 implies collapse not only of the transformed mean velocity but also of semilocal eddy viscosity and TKE production. Existing compressibility transformations are shown, using hypersonic DNS, to incur velocity errors of 1-25% relative to the incompressible inner-outer model, particularly for strongly cooled cases. A new forward compressible-to-incompressible transformation is 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 and wall heat transfer rate. Casewise optimization yields consistency errors of 1-4% across the available hypersonic DNS set, and this performance is retained using multi-linear and multi-quadratic regressions. The forward transformation is embedded in an inverse incompressible-to-compressible transformation framework, which reconstructs the compressible state from freestream and wall conditions at a prescribed BL thickness. The inverse solver recovers key BL 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 TBLs with strong wall cooling.