Excitation and stability of nonlinear compressible G\"ortler vortices and streaks induced by free-stream vortical disturbances

TL;DR

This study investigates the nonlinear development and stability of G"ortler vortices and streaks in compressible boundary layers influenced by free-stream vortical disturbances, with implications for turbine blade flows and transition to turbulence.

Contribution

It provides a comprehensive analysis of vortex formation, stability, and secondary instabilities in compressible flows over curved surfaces, including new modes and detailed parametric effects.

Findings

Increased G"ortler number enhances boundary layer instability.

Higher Mach number or disturbance frequency stabilizes the flow.

Identified new varicose even mode promoting transition.

Abstract

We study the generation, nonlinear development and secondary instability of unsteady G\"ortler vortices and streaks in compressible boundary layers exposed to free-stream vortical disturbances and evolving over concave, flat and convex walls. The formation and evolution of the disturbances are governed by the compressible nonlinear boundary-region equations, supplemented by initial and boundary conditions that characterise the impact of the free-stream disturbances on the boundary layer. Computations are performed for parameters typical of flows over high-pressure turbine blades, where the G\"ortler number, a measure of the curvature effects, and the disturbance Reynolds number, a measure of the nonlinear effects, are order-one quantities. At moderate intensities of the free-stream disturbances, increasing the G\"ortler number renders the boundary layer more unstable, while increasing the Mach number or the frequency stabilises the flow. As the free-stream disturbances become more intense, vortices over concave surfaces no longer develop into the characteristic mushroom-shaped structures, while the flow over convex surfaces is destabilised. An occurrence map identifies G\"ortler vortices or streaks for different levels of free-stream disturbances and G\"ortler numbers. Our calculations capture well the experimental measurements of the enhanced skin friction and wall-heat transfer over turbine-blade pressure surfaces. The time-averaged wall-heat transfer modulations, termed hot fingers, are elongated in the streamwise direction and their spanwise wavelength is half of the characteristic wavelength of the free-stream disturbances. Nonlinearly saturated disturbances are unstable to secondary high-frequencymodes, whose growth rate increases with the G\"ortler number. A new varicose even mode is reported, which may promote transition to turbulence at the stem of nonlinear streaks.