Beyond secondary instability: on the emergence of finite-amplitude waves in G\"ortler vortices

TL;DR

This paper introduces a novel computational approach using the PCS method to accurately predict the finite-amplitude waves in G"ortler vortices, advancing understanding of transition to turbulence.

Contribution

The study applies the PCS method to G"ortler vortices, capturing nonlinear wave evolution and matching experimental observations, thus improving predictive capabilities.

Findings

Successfully reproduces wave amplitude observed in experiments.

Accurately predicts displacement thickness in G"ortler vortices.

Demonstrates the effectiveness of PCS in nonlinear vortex-wave interactions.

Abstract

G\"ortler vortices developing over a concave wall support rapidly oscillating wavelike disturbances through secondary instabilities. Although experiments indicate that the finite-amplitude evolution of these waves acts as a precursor to turbulence transition, accurate and efficient prediction has remained out of reach. We overcome this limitation by using the Parabolised Coherent Structures (PCS) method of Song & Deguchi (2025), which incorporates the nonlinear vortex-wave interaction into a standard spatial-marching approach. Our computations successfully reproduce the wave amplitude and displacement thickness observed in the widely known experiments of Swearingen & Blackwelder (1987).