A Three-Dimensional Two-Temperature Gas-Kinetic Scheme with Generalized Kinetic Boundary Condition for Hypersonic SBLI

Accurate prediction of aerothermal loads in hypersonic flows is critical yet challenging due to the coupling of Shock-Wave/Boundary-Layer Interactions (SBLI) and thermal non-equilibrium. This work presents the development of a three-dimensional two-temperature Gas-Kinetic Scheme (3D 2T-GKS) on unstructured meshes. The scheme resolves translational-rotational and vibrational energy modes within a unified kinetic framework. A key innovation is the integration of a Generalized Kinetic Boundary Condition (GKBC), which physically decouples the thermal accommodation of vibrational energy from the translational-rotational mode, thereby offering a more accurate model for gas-surface interactions. Additionally, a Discontinuity Feedback Factor (DFF) is employed to capture strong shock waves with reduced numerical dissipation compared to classical limiters. The method is rigorously validated against standard experimental benchmarks, including the sharp double-cone and hollow cylinder-flare configurations. Numerical results demonstrate that the proposed solver, augmented by the GKBC, accurately captures complex wave structures, separation topologies, and surface heat flux distributions. These findings confirm the robustness and fidelity of the 3D 2T-GKS for simulating complex hypersonic non-equilibrium flows.

💡 Research Summary

The paper presents a comprehensive three‑dimensional two‑temperature gas‑kinetic scheme (3D 2T‑GKS) designed to accurately predict aerothermal loads in hypersonic flows where shock‑wave/boundary‑layer interactions (SBLI) are tightly coupled with thermal non‑equilibrium. Traditional Navier‑Stokes (NS) solvers, even when augmented with multi‑temperature models, rely on near‑equilibrium assumptions and linear constitutive relations, which become invalid in strong shock layers and near‑wall Knudsen regions. Moreover, conventional finite‑volume implementations evaluate surface heat flux from the first cell’s macroscopic state, making predictions highly sensitive to the first‑cell height and incapable of representing the non‑equilibrium distribution function at the gas‑solid interface, especially the slow accommodation of vibrational energy.

To overcome these limitations, the authors formulate a BGK‑type kinetic model that distinguishes translational‑rotational temperature (T_tr) from vibrational temperature (T_v). The collision term is split into an elastic relaxation toward an intermediate equilibrium f_eq (characterized by T_tr for translational‑rotational modes and T_v for vibrational modes) and an inelastic relaxation toward the full Maxwellian g. The vibrational collision number Z_v, derived from a modified Millikan‑White expression, controls the timescale of vibrational‑translational energy exchange. By performing a Chapman‑Enskog expansion, the corresponding macroscopic equations are obtained: continuity, momentum, total‑energy, and a separate vibrational‑energy equation with a source term s = (ρE_v)^eq – ρE_v / (Z_v τ). Viscous stresses and heat fluxes contain contributions from both temperature fields, providing mode‑specific transport coefficients.

A central innovation is the Generalized Kinetic Boundary Condition (GKBC). Unlike the classical Maxwell slip‑jump condition, which ties momentum and thermal accommodation coefficients together and forces all internal modes to equilibrate at the wall at the same rate, GKBC decouples these processes by employing particle‑scattering kernels that assign distinct accommodation coefficients to translational‑rotational and vibrational energies. This physically realistic treatment eliminates the systematic over‑prediction of surface heat flux observed in conventional models, especially in low‑density, high‑temperature regimes where vibrational modes relax much more slowly.

To capture strong shocks with minimal numerical dissipation, the authors introduce a Discontinuity Feedback Factor (DFF). DFF acts as an adaptive limiter that reduces artificial viscosity in smooth regions while providing sufficient dissipation near discontinuities, thereby preserving sharp shock structures without inducing spurious oscillations.

The method is implemented on unstructured meshes within a finite‑volume framework. Validation is performed against two canonical hypersonic SBLI experiments conducted at the LENS shock tunnel: the sharp double‑cone and the hollow cylinder‑flare configurations. For both cases, the 3D 2T‑GKS with GKBC reproduces pressure distributions, surface heat‑flux profiles, and separation‑bubble extents with errors typically below 5 % relative to measurements. Notably, the inclusion of GKBC markedly improves the prediction of vibrational temperature fields and surface heat flux, correcting the over‑predictions seen when a standard Maxwell boundary condition is used. A further test on an asymmetric double‑cone at a 2° angle of attack demonstrates the scheme’s capability to resolve three‑dimensional, non‑symmetric SBLI and re‑attachment phenomena, which are missed by conventional NS‑based multi‑temperature solvers.

The authors conclude that the integrated 3D 2T‑GKS, GKBC, and DFF framework provides a robust, high‑fidelity tool for hypersonic aerothermodynamics. It overcomes the intrinsic limitations of NS‑based approaches, delivers accurate wall‑heat‑flux predictions without excessive mesh refinement, and is readily extensible to include chemical reactions, multi‑species gases, and more complex gas‑surface interaction models. This advancement holds significant promise for the design and analysis of next‑generation hypersonic vehicles where reliable prediction of coupled SBLI and thermal non‑equilibrium is essential.