Beyond classical similitude: group theoretic extrapolation of hypersonic stagnation-point boundary layers

Abstract: Motivated by the need to extrapolate the results from ground-based experiments to the conditions of high-speed flight, we present the Lie equivalence symmetry analysis of the hypersonic stagnation-point boundary layers. We demonstrate the application of the equivalence symmetry on the set of coupled ODEs which are physically relevant in hypersonics. By allowing the property laws to transform along with the independent and dependent variables, the invariants derived within the similarity-reduced stagnation-point ODE formulation, identify families of non-linear maps that can be used to extrapolate the laboratory-scale predictions to flight. In practice, implementing these maps requires the laboratory-scale ODE solution together with the lab- and flight-side thermochemical property data, which are generally available from existing databases. The maps are derived for the similarity-reduced non-dimensional temperature across the boundary layer and are shown to {collapse with independently computed flight solutions} for a range of relevant cases.

• The paper presents a novel Lie equivalence symmetry framework that generalizes hypersonic boundary layer extrapolation beyond classical similitude.
• It derives nonlinear mappings using invariants from thermochemical property laws to accurately correlate laboratory measurements with flight conditions.
• Numerical validations demonstrate robust near-wall heat-flux agreement while quantifying deviations in the outer boundary layer for improved extrapolation fidelity.

Group-Theoretic Extrapolation for Hypersonic Stagnation-Point Boundary Layers

Introduction

The persistent gap between laboratory and true flight conditions in hypersonic flows, particularly for stagnation-point boundary layers, fundamentally constrains aerothermodynamic extrapolation. Classically, similarity rules such as binary scaling and Local Heat Transfer Simulation (LHTS) are used to extrapolate laboratory measurements to flight environments. However, their efficacy is limited due to the inability of ground-based facilities to simultaneously match the thermochemical, flow, and material response variables that dominate hypersonic regimes. This paper addresses these limitations by leveraging Lie equivalence symmetries to generalize the extrapolation process beyond traditional similitude, introducing a new formalism that enables nonlinear mappings of physical solutions across arbitrary thermochemical property laws.

Mathematical Framework: Lie Equivalence Symmetries

The authors apply Lie equivalence symmetry analysis to the similarity-reduced system of ODEs governing stagnation-point boundary-layer flows. Unlike classical Lie symmetries, which act on the differential equation's field variables, Lie equivalence symmetries incorporate transformations of both the independent/dependent variables and the arbitrary constitutive property laws (e.g., viscosity, conductivity, specific heats). This generalization permits the construction of invariants and nonlinear mappings that relate entire classes of problems rather than isolated solutions.

The paper details the derivation of infinitesimal generators and their prolongations for the governing ODE system, projecting the universal generator onto auxiliary and primary equations to obtain the required symmetry relations. The resulting invariants underpin a nonlinear mapping for the reduced temperature profile $g = T/T_e$ across the boundary layer, where $T_e$ is the edge temperature. The analysis yields determining equations that constrain the allowable symmetry transformations explicitly in terms of the arbitrary property data.

Integral Invariant and Mapping Procedure

A key theoretical result is the formulation of an integral invariant built from thermochemical property tables, primarily the thermal conductivity $\chi(g)$, which is numerically inverted to establish a temperature map between laboratory and flight conditions. The mapping is formalized as:

$g^\ast = H_F^{-1}\left( e^K H_L(g) \right)$

where $H_L$ and $H_F$ denote integrals of $\chi$ with respect to laboratory and flight data, respectively, $K$ is determined through boundary matching (typically at the wall), and $g^\ast$ is the mapped non-dimensional temperature. This invariant is strictly monotone, ensuring invertibility and robust numerical implementation, leveraging high-temperature databases such as Mutation++ for tabulated property laws.

Pragmatically, only the laboratory profile and property tables for both lab and flight conditions are required. No explicit knowledge of the flight temperature profile as a function of wall-normal coordinate is necessary.

Numerical Results and Validation

Numerical tests are conducted on three distinct boundary-layer configurations, varying wall and edge temperatures between laboratory and flight cases. Using PlasFlowSolver (Lanza et al. 2025), solutions to the ODEs are computed for a complex air mixture, including dissociative and high-enthalpy effects. The integral invariants are numerically evaluated and inverted using discrete interpolation, matching wall heat-flux, edge states, and stagnation pressure where possible.

The mapped laboratory temperature profile $\hat{g}$ demonstrates a strong collapse with the independently computed flight-side ODE solution $T_e$0 across the boundary layer, particularly in the inner region ($T_e$1). This holds even when traditional LHTS criteria are not identically matched, highlighting the robustness of the group-theoretic mapping. Deviations appear primarily in the outer boundary layer, attributed to failure of invariant conditions and loss of parameter compatibility, but near-wall heat transfer and gradients are maintained with high fidelity.

Other property laws such as specific heats $T_e$2 are comparably compatible with the $T_e$3-inferred map, though exact image matching is only theoretically guaranteed for members of the equivalence class. The physical property data can deviate from the transformed laboratory counterparts, but the Lie equivalence map quantifies these discrepancies, providing a rigorous measure of extrapolation quality.

Implications and Future Directions

By constructing group-theoretic mappings, this study advances the theoretical foundation of hypersonic lab-to-flight extrapolation, allowing predictive modeling beyond the constraints of classical similitude. The framework obviates the need for simultaneous matching of all nondimensional control parameters, enabling systematic treatment of complex aerothermochemical phenomena (including dissociation, radiation, and multi-species effects) using only property law data and laboratory profiles.

Theoretical implications suggest that classical similarity parameters are subsets of broader group invariants, and that identity mapping (as in LHTS) is merely a degenerate case. Practically, the Lie equivalence approach facilitates accurate near-wall heat-flux prediction and boundary-layer reparametrization, crucial for flight performance assessment, heat-shield design, and risk mitigation in atmospheric re-entry and high-speed vehicles.

Future research directions include extension of the mathematical framework to fully multidimensional PDE systems, inclusion of additional physical complexities (chemical/thermal nonequilibrium, radiation), and validation against high-fidelity CFD and experimental datasets. Enhanced numerical integration, machine-aided property law tabulation, and uncertainty quantification in extrapolation will further streamline practical application to flight-test campaigns and facility design.

Conclusion

The paper establishes a rigorous group-theoretic foundation for extrapolation-to-flight in hypersonic stagnation-point boundary layers, superseding classical similitude with Lie equivalence invariants that enable nonlinear, property-law-driven mappings. Numerical implementations confirm robust near-wall agreement between laboratory and flight solutions, revealing departures in outer layers that yield precise measures of extrapolation fidelity. The analysis provides both a proof of concept and an extensible framework, setting the stage for more generalized and comprehensive approaches in hypersonic aerothermodynamics.