First-order buoyancy correction of modal instabilities in stratified boundary layers

We present a perturbation-based framework that captures buoyancy effects on modal instabilities in stratified boundary-layer flows within the fully compressible, non-Oberbeck–Boussinesq formulation. Treating the Richardson number as a small parameter and recasting the stability problem into an adjoint-residual form, we derive a first-order correction for the eigenvalues using only the neutrally buoyant eigenvalue problem. The framework applies to both ideal-gas and non-ideal fluid boundary layers and eliminates the need to re-solve the eigenvalue problem at each stratification level at minimal computational cost. For ideal-gas boundary layers, the framework accurately predicts how stable and unstable stratification modifies Tollmien–Schlichting waves, from growth rates and eigenfunctions to $N$ -factors, across a wide range of Prandtl numbers, temperature ratios and Mach numbers. Notably, the buoyancy sensitivity varies strongly with Prandtl number, revealing that for a given Richardson number, buoyancy can switch from destabilising to stabilising depending on the fluid. Beyond ideal-gas conditions, we apply the first-order buoyancy correction to strongly stratified boundary layers with supercritical fluids, where the phase relationship between density and velocity perturbations determines whether buoyancy stabilises or destabilises the underlying instability. The resulting $N$ -factors demonstrate, for the first time, that buoyancy significantly affects transition predictions under pseudo-boiling conditions.