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

TL;DR

This paper develops a first-order correction framework for buoyancy effects on modal instabilities in stratified boundary layers, enabling efficient stability analysis across various fluid conditions without re-solving eigenvalue problems.

Contribution

It introduces a perturbation-based method that accurately predicts buoyancy influence on flow stability in stratified boundary layers, applicable to ideal gases and supercritical fluids.

Findings

Buoyancy effects vary strongly with Prandtl number.

Buoyancy can switch from destabilising to stabilising depending on fluid properties.

Buoyancy significantly impacts transition predictions under pseudo-boiling conditions.

Abstract

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. This eliminates the need to re-solve the eigenvalue problem at each stratification level. 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, holding 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.