Determine whether the collective displacement persists at higher Knudsen numbers

Determine whether a collective bow-layer displacement of amplitude comparable to that resolved at Kn_D=0.01 and 0.025 persists for 0.05≤Kn_D≤0.15, using measurements with sufficient sensitivity to distinguish persistence from physical disappearance.

Background

The analysis robustly resolves a same-signed, slowly varying angular displacement at Kn_D=0.01 and 0.025. At higher Knudsen numbers, however, the available records have coarser temporal sampling and substantially poorer detectability. Injection tests show that the existing data cannot reliably detect a mode with the low-Knudsen reference amplitude in much of the range 0.05≤Kn_D≤0.15.

Consequently, the absence of a classified mode at higher Knudsen numbers cannot be interpreted as evidence that the collective displacement physically disappears or that a critical Knudsen number exists. Longer records, finer output cadence, increased particle loading, or otherwise more sensitive measurements are needed to resolve this question.

References

Consequently, the records over 0.05 ≤ Kn_D ≤ 0.15 cannot exclude persistence of a low-Knudsen-sized mode. The result is resolution at two near-continuum states, not demonstrated disappearance with rarefaction.

— Noise-separated evidence for a slow collective displacement in a rarefied hypersonic bow-shock layer  (2608.17285 - Shoja-sani et al., 18 Aug 2026) in Section 3.9, “What can be concluded beyond Kn_D=0.025”; Section 4.7, “Limitations and scope of the conclusions”