Spatial amplification owing to modal instability plays a significant role in determining when and how a hypersonic boundary layer transitions from a laminar state to turbulence. Traditional stability analysis methods rely on the strong assumption of a slowly-varying baseflow, which limits their predictive power. Such methods can calculate downstream amplification only relative to upstream points where the flow has already passed through shock waves (such as a vehicle’s bow shock) and that are far away from complex geometry (such as a blunt nose tip). The estimation of perturbation amplitudes at these upstream points from freestream disturbance levels then requires empirical formulae, calibrated for different geometries on a case-by-case basis. As a further complication, hypersonic boundary layers also support non-modal amplification mechanisms not captured by traditional stability analysis.
We instead investigate this problem using Input/Output (I/O) analysis which directly relates freestream perturbations to the total system response through the resolvent operator, eliminating the need for a slowly-varying base flow. As a global method, I/O analysis simultaneously incorporates receptivity, modal amplification, and non-modal amplification mechanisms. Furthermore, as a decomposition method, I/O analysis provides a technique to distinguish between different mechanisms competing in a single flow. Crucial to the accurate representation of receptivity, our method employs a linear model for the transmission and reflection of small perturbations through a shock wave. We validate our method through comparison to schlieren measurements of hypersonic instabilities over ogive-cylinders taken in the AFRL Mach-6 Ludwieg Tube. In addition to the expected Mack 2nd mode instability, I/O analysis predicts a new type of modal instability, as well as non-modal entropy-layer instability, in good agreement with experimental observations. The interaction between these different mechanisms may explain the phenomenon of “transition reversal” with increasing nose-tip bluntness.