Bulk-to-Wall Observability in Rarefied Hypersonic Flow: Moment Limits and Incident Half-Range Sufficiency

Abstract

Wall traction in a rarefied gas is a half-range functional of molecules arriving at and leaving a surface, whereas hybrid solvers, moment methods, and learned wall models often transmit only finite full-range moments. We determine which directional information is lost and what restores the wall functional using direct simulation Monte Carlo (DSMC) of Mach-6 flow over three triangular-protrusion orientations. Output-specialized neural networks interpolate a 57-condition DSMC database. A distinct ExtraTrees regression holds learner capacity, auxiliary features, train/test split, and seeds fixed while comparing the primitive state $S_0$, the momentum-flux-augmented state $S_1$, and the heat-flux-augmented state $S_2$. Adding the momentum-flux tensor $P_{ij}$ reduces aggregate pressure error by 31\% but does not make signed shear transferable. An analytic null-space construction supplies the mechanism: strictly positive distributions can share all full-range moments through degree three while producing different pressure and opposite signed shear because the wall kernel selects the incident half space. A finite-distance off-wall ExtraTrees state $S_{\mathrm{off}}$ shows that partial directionality is useful but representation-dependent. Finally, a parameter-free kinetic reconstruction combines the incident half-range wall-arrival state $S_{\mathrm{HR}}$ with the prescribed diffuse kernel and agrees with both a same-window DSMC tally sharing the incident events and a separate 40,000-step DSMC wall-tally window. The resulting design principle is direct: a rarefied wall closure must preserve the incident normal--tangential correlations required by the target load rather than merely adding higher full-range moments.