On a slender hypersonic body (M ≳ 4–5), the dominant instability of the laminar boundary layer is the Mack second mode — an acoustic wave trapped between the wall and the boundary-layer edge, with wavelength ≈ 2δ (δ = local boundary-layer thickness). Engineers predict transition with the eN method: integrate the local disturbance growth rate downstream into an amplification factor N(x), and call the flow turbulent once N(x) crosses a critical value Ntr set by the "noisiness" of the environment (wind-tunnel acoustic noise or in-flight surface roughness):
N(x) = ∫ σ(x') dx' transition when N(x) = N_tr
This simulator uses the simplified engineering scaling
N(x) ≈ K(M, T_w) · √(Re_x / 10⁶), Re_x = (unit Re) · x
K grows with (M − 3.5) — second mode strengthens with Mach —
and with wall cooling (1 − T_w/T_aw): a cold wall is a
well-documented destabilizer of the second mode even
though it stabilizes the older first (Tollmien–Schlichting) mode.
- Mach slider — raises K(M,Tw), so the same station reaches higher N sooner and transition moves toward the nose.
- Wall cooling Tw/Taw — a colder wall (small Tw/Taw, typical with active thermal-protection cooling) amplifies second-mode growth and moves transition forward.
- Unit Reynolds number — sets how fast the local Rex = (unit Re)·x builds along the body; a denser/faster flow (higher altitude aside, higher unit Re) reaches the critical N sooner in physical distance.
- Environment N-factor threshold — a "quiet" flight or low-disturbance tunnel needs Ntr ≈ 9–10 to trip; a noisy ground tunnel or a rough/tiled surface effectively lowers Ntr toward 4–6, triggering earlier ("bypass") transition.
Tracer particles ride the boundary-layer surface: blue = laminar, amber = an amplifying second-mode wave packet as N approaches Ntr, red = broken-down turbulent flow past the transition front. The side graph plots N(x) against the dashed Ntr threshold, with a marker at the predicted transition point — the same eN chart used to design real hypersonic vehicles (e.g. HIFiRE, X-51) and their thermal-protection systems, since transition can multiply local heat flux several-fold. This is a simplified, illustrative scaling law, not a substitute for full linear-stability-theory (LST) computations.