On Continuous Spectra of the Orr-Sommerfeld/Squire Equations and Entrainment of Free-stream Vortical Disturbances in the Blasius Boundary Layer

Abstract

Small-amplitude perturbations are governed by the linearized Navier-Stokes equations, which are, for a parallel or nearly parallel shear flow, customarily reduced to the OrrSommerfeld (O-S) and Squire equations. In this paper, we consider continuous spectra (CS) of the O-S and Squire operators for the Blasius boundary layer, and address the issue of whether and when continuous modes can represent free-stream vortical disturbances and their entrainment into the shear layer. We highlight two particular properties of the CS: (a) the eigenfunction of a continuous mode simultaneously consists of two components with wall-normal wavenumbers ±k2, a phenomenon which we refer to as ‘entanglement of Fourier components’; and (b) for low-frequency disturbances the presence of the boundary layer forces the streamwise velocity in the free stream to take a much larger amplitude than those of the transverse velocities. Both features appear to be non-physical, and cast some doubt about the appropriateness of using CS to characterize free-stream vortical disturbances and their entrainment into the boundary layer, a practice that has been adopted in some recent studies of bypass transition. A high-Reynolds-number asymptotic description of continuous modes and entrainment is present, and it shows that the entanglement is a result of neglecting non-parallelism, which has a leading-order effect on the entrainment. When this effect is included, entanglement disappears, and moreover the streamwise velocity is significantly amplified in the edge layer when R ω 1, where R is the Reynolds number based on the local boundary-layer thickness.