Tollmien–Schlichting waves in the subsonic regime

Abstract

The Tollmien–Schlichting (T-S) waves play a key role in the early stages of boundary layer transition. In a breakthrough work, Grenier, Guo, and Nguyen gave the first rigorous construction of the T-S waves of temporal mode for the incompressible fluid. Yang and Zhang recently made an important contribution by constructing the compressible T-S waves of temporal mode for certain boundary layer profiles with Mach number . In this paper, we construct the T-S waves of both temporal mode and spatial mode to the linearized compressible Navier–Stokes system around the boundary layer flow in the whole subsonic regime , including the Blasius profile. Our approach is based on a novel iteration scheme between the quasi-incompressible and quasi-compressible systems, with a key ingredient being the solution of an Orr–Sommerfeld-type equation using a new Airy–Airy–Rayleigh iteration instead of Rayleigh–Airy iteration introduced by Grenier, Guo, and Nguyen. We believe that the method developed in this work can be applied in solving other related problems for subsonic flows.