Predictive Compressibility Transformation for Hypersonic Turbulent Boundary Layers with Cold Walls

TL;DR

The paper addresses the challenge of mapping compressible hypersonic ZPG TBLs to incompressible references using a strict, Mach-independent consistency criterion: a single incompressible inner–outer mean-shear model must reproduce all transformed profiles when expressed in a transformed wall-normal coordinate. It introduces a forward compressible-to-incompressible transformation based on a convex combination of semilocal and integral basis functions with coefficients that depend on the friction Mach number $M_\tau$ and wall heat transfer rate $B_q$, achieving $1$–$4\%$ consistency across a hypersonic DNS set, and embeds this in an inverse incompressible-to-compressible framework to reconstruct compressible fields from freestream and wall data. The forward model substantially improves upon existing transformations (GFM, Volpiani, HLLP) by enforcing a consistent inner–outer representation, while the inverse solver accurately recovers BL parameters, velocity, and wall quantities, especially in cold-wall cases, and shows better performance than the inverse HLLP model. The proposed framework provides a physically grounded basis for near-wall modeling in hypersonic TBLs and has implications for heat-transfer predictions and wall-modeled LES in high-speed flows.

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.

Figures (8)

• Figure 1: Maximum relative error $\varepsilon_{\text{model}}$ of the incompressible inner--outer model \ref{['eq:F_inner_outer']} compared with DNS schlatter2010 and LES schlatterLES2014 reference data.
• Figure 2: Workflow of the inverse (incompressible-to-compressible) transformation.
• Figure 3: Consistency assessment of existing compressibility transformations with respect to the incompressible inner--outer model in \ref{['sec:consistency']}. Solid lines show the incompressible inner--outer model prediction $U_{\text{model}}^+(Y^+)$, and dotted lines show the transformed profiles $U_{\text{transformed}}^+(Y^+)$. Colors denote different hypersonic ZPG TBL cases in Zhang2018: M2p5, M6Tw025, M6Tw076, M8Tw048, and M14Tw018.
• Figure 4: Transformed velocity profiles obtained with the proposed forward transformation using (a) the multi-linear and (b) the multi-quadratic fit in $(M_\tau,B_q)$. Solid lines show the incompressible inner--outer model $U_{\text{model}}^+(Y^+_{\text{inc}})$, and dotted lines show the transformed profiles $U_{\text{transformed}}^+(Y^+_\text{inc})$ with the corresponding regression-based coefficients. Colors denote different hypersonic ZPG TBL cases in Zhang2018: M2p5, M6Tw025, M6Tw076, M8Tw048, and M14Tw018.
• Figure 5: Inverse reconstruction of mean velocity profiles at the DNS sampling station.