Physics-Based Machine Learning Closures and Wall Models for Hypersonic Transition-Continuum Boundary Layer Predictions

TL;DR

This work tackles the difficulty of predicting nonequilibrium hypersonic transition–continuum flows where traditional continuum models falter. It introduces physics-constrained ML closures embedded in the compressible Navier–Stokes equations, trained with adjoint-based optimization to ensure consistency with governing physics, and couples them with a novel wall model based on skewed-Gaussian velocity distributions. The study develops three transport-closure families (Isotropic, Anisotropic, Anisotropic–TF) and a Boltzmann-distribution-wall model, demonstrating substantial accuracy gains over DSMC benchmarks for 2D flat-plate flows and improved extrapolation across Mach and Knudsen numbers, especially when trained in parallel across multiple conditions. The results show notable improvements in wall-flux predictions and near-wall dynamics, while maintaining computational efficiency (orders of magnitude faster than DSMC) and stability. This framework paves the way for reliable, data-driven continuum modeling in regimes where kinetic effects are pronounced, with potential extensions to chemistry, multicomponent flows, and more complex geometries.

Abstract

Modeling rarefied hypersonic flows remains a fundamental challenge due to the breakdown of classical continuum assumptions in the transition-continuum regime, where the Knudsen number ranges from approximately 0.1 to 10. Conventional Navier-Stokes-Fourier (NSF) models with empirical slip-wall boundary conditions fail to accurately predict nonequilibrium effects such as velocity slip, temperature jump, and shock structure deviations. We develop a physics-constrained machine learning framework that augments transport models and boundary conditions to extend the applicability of continuum solvers in nonequilibrium hypersonic regimes. We employ deep learning PDE models (DPMs) for the viscous stress and heat flux embedded in the governing PDEs and trained via adjoint-based optimization. We evaluate these for two-dimensional supersonic flat-plate flows across a range of Mach and Knudsen numbers. Additionally, we introduce a wall model based on a mixture of skewed Gaussian approximations of the particle velocity distribution function. This wall model replaces empirical slip conditions with physically informed, data-driven boundary conditions for the streamwise velocity and wall temperature. Our results show that a trace-free anisotropic viscosity model, paired with the skewed-Gaussian distribution function wall model, achieves significantly improved accuracy, particularly at high-Mach and high-Knudsen number regimes. Strategies such as parallel training across multiple Knudsen numbers and inclusion of high-Mach data during training are shown to enhance model generalization. Increasing model complexity yields diminishing returns for out-of-sample cases, underscoring the need to balance degrees of freedom and overfitting. This work establishes data-driven, physics-consistent strategies for improving hypersonic flow modeling for regimes in which conventional continuum approaches are invalid.

Figures (25)

• Figure 1: Schematic of the flat plate geometry showing domain dimensions and boundary conditions.
• Figure 2: Schematic of the neural network architecture used for augmented transport models.
• Figure 3: Loss convergence of a co-optimized Anisotropic-TF transport model and distribution function-based wall model for a $M_\infty = 7$ flat-plate boundary layer.
• Figure 4: Streamwise velocity distribution functions obtained from DSMC, the Maxwell--Boltzmann distribution function, and the modeled skewed-Gaussian distribution function at wall locations $x = 0.05$ m and $x = 0.55$ m for a $M_\infty = 7$ flat-plate boundary layer.
• Figure 5: Local density gradient-based Knudsen number for freestream conditions $U_{\infty} = 2457.16$ m/s, $T_{\infty}=300$ K and $\rho_{\infty} = 9.89013^{-5}$ kg/$m^3$, showing continuum breakdown ($\mathrm{Kn}=0.1$ isocontours) behind the shock and in the boundary layer.