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Quasi-modes in boundary-layer-type flows. Part 1. Inviscid two-dimensional spatially harmonic perturbations

  • VICTOR I. SHRIRA (a1) (a2) and IGOR A. SAZONOV (a1)

Abstract

The work, being the first in a series concerned with the evolution of small perturbations in shear flows, studies the linear initial-value problem for inviscid spatially harmonic perturbations of two-dimensional shear flows of boundary-layer type without inflection points. Of main interest are the perturbations of wavelengths 2π/k long compared to the boundary-layer thickness H, kH = ε [Lt ] 1. By means of an asymptotic expansion, based on the smallness of ε, we show that for a generic initial perturbation there is a long time interval of duration ∼ ε−3 ln(1/ε), where the perturbation representing an aggregate of continuous spectrum modes of the Rayleigh equation behaves as if it were a single discrete spectrum mode having no singularity to the leading order. Following Briggs et al. (1970), who introduced the concept of decaying wave-like perturbations due to the presence of the ‘Landau pole’ into hydrodynamics, we call this object a quasi-mode. We trace analytically how the quasi-mode contribution to the entire perturbation field evolves for different field characteristics. We find that over O−3 ln(1/ε)) time interval, the quasi-mode dominates the velocity field. In particular, over this interval the share of the perturbation energy contained in the quasi-mode is very close to 1, with the discrepancy in the generic case being O4) (O4) for the Blasius flow). The mode is weakly decaying, as exp(−ε3t). At larger times the quasi-mode ceases to dominate in the perturbation field and the perturbation decay law switches to the classical t−2. By definition, the quasi-modes are singular in a critical layer; however, we show that in our context their singularity does not appear in the leading order. From the physical viewpoint, the presence of a small jump in the higher orders has little significance to the manner in which perturbations of the flow can participate in linear and nonlinear resonant interactions. Since we have established that the decay rate of the quasi-modes sharply increases with the increase of the wavenumber, one of the major conjectures of the analysis is that the long-wave components prevail in the large-time asymptotics of a wide class of initial perturbations, not necessarily the predominantly long-wave perturbations. Thus, the explicit expressions derived in the long-wave approximation describe the asymptotics of a much wider class of initial conditions than might have been anticipated. The concept of quasi-modes also enables us to shed new light on the foundations of the method of piecewise linear approximations widely used in hydrodynamics.

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Quasi-modes in boundary-layer-type flows. Part 1. Inviscid two-dimensional spatially harmonic perturbations

  • VICTOR I. SHRIRA (a1) (a2) and IGOR A. SAZONOV (a1)

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