Quasi-modes in boundary-layer-type flows. Part 2. Large-time asymptotics of broadband inviscid small-amplitude two-dimensional perturbations

摘要

The paper is the second in a series concerned with the development of the quasimode concept in the context of boundary layers. The evolution of localized two-dimensional perturbations in boundary layers without inflection points is considered within the framework of linear inviscid theory. Making use of the results of Part 1 (Shrira & Sazonov 2001) for monochromatic perturbations it is shown that arbitrary broadband initial perturbations tend to a universal asymptotic regime at large times t. We refer to the phenomenon of the emergence of the asymptotic regime as 'adjustment'. The regime itself corresponds to a slow dynamics of long-wave triple-deck-type perturbations and is described well as the evolution of a single quasimode, which allows dramatic simplification of its description. At the asymptotic stage the spatio-temporal structure of the perturbation is explicitly described in terms of Fresnel's functions with coefficients specified by certain integrals of the initial distribution. Asymptotically the perturbation represents a sharply localized group of decaying oscillations which propagate with celerity approaching the mean flow velocity at the surface. For generic perturbations, the decay, in terms of streamwise velocity, is t(-1/2). The envelope of the group is formed by the Landau damping intrinsic to the quasi-modes, and the length of the group and the number of oscillations in the group grow with time as t(2/3) and t(1/6), respectively. The evolution of the non-quasi-modal part is also investigated. The vorticity perturbation is found to form a vortex patch shaped like a comet tail and advected by the mean flow. The picture of evolution established for generic perturbations is found to hold for several classes of non-generic but physically relevant initial distributions; the corresponding solutions are presented and discussed. The analytical results have been confirmed by the direct numerical simulation of the linearized primitive equations. [References: 26]