On vortical disturbances in single and two-fluid boundary layers

Abstract

The vortical disturbance environment surrounding a laminar boundary layer affects the proceedings of transition to turbulence. The exponentially growing Tollmien–Schlichting wave is observed under low free-stream turbulence levels, and is replaced by the algebraically growing streaks upon further increase in the turbulence intensity. This scenario is significantly affected by the presence of wall films which introduce additional instabilities and alter the amplification of the streaks. In this work, the effect of wall films on the linear stability of boundary layers is investigated using the Orr-Sommerfeld, Squire and the interface displacement equations. A modal analysis is conducted first, in order to identify all the unstable modes and their respective regimes of dominance. Furthermore, the physical mechanisms contributing to disturbance growth are studied using the kinetic energy equation. No unstable eigenvalues are found for wall films less viscous than the outer stream. Under such conditions, the streaks are likely to dominate; their amplification being dependent on the penetration of the free-stream vortical disturbances into the boundary layer. The ingestion of the free-stream vorticity in the mean shear is explained using the continuous spectrum of the Orr-Sommerfeld equation. A unique parameter is identified to distinguish three asymptotic regimes representing complete, partial and negligible penetration into the boundary layer, respectively. The physical mechanism is a competition between viscous diffusion and convection by the mean flow. The wall film affects the penetration into the boundary layer by modifying the wall-normal wavenumber across the interface. The penetrating free-stream disturbances efficiently generate streaks by tilting the mean vorticity. Their amplification is investigated using an initial value problem that describes the evolution of a linear perturbation. Lower viscosity wall films reduce the amplification of the streaks. However, another growth mechanism arising from the interfacial displacement dominates at long time and is enhanced for lower viscosity films.