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

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 o f boundary-layer type without inflection points. Of main interest are the p erturbations of wavelengths 2 pi /k long compared to the boundary-layer thi ckness H, kH = epsilon much less than 1. By means of an asymptotic expansio n, based on the smallness of e, we show that for a generic initial perturba tion there is a long time interval of duration similar to epsilon (-3) ln(l /epsilon), where the perturbation representing an aggregate of continuous s pectrum modes of the Rayleigh equation behaves as if it were a single discr ete spectrum mode having no singularity to the leading order. Following Bri ggs et al. (1970), who introduced the concept of decaying wave-like perturb ations due to the presence of the 'Landau pole' into hydrodynamics, we call this object a quasi-mode. We trace analytically how the quasi-mode contrib ution to the entire perturbation field evolves for different field characte ristics. We find that over O(epsilon (-3) ln(l/e)) time interval, the quasi -mode dominates the velocity field. In particular, over this interval the s hare, of the perturbation energy contained in the quasi-mode is very close to 1, with the discrepancy in the generic case being O(epsilon (4)) (O(epsi lon (6)) for the Blasius flow). The mode is weakly decaying, as exp(-epsilo n (3)t). At larger times the quasi-mode ceases to dominate in the perturbat ion field and the perturbation decay law switches to the classical t(-2). B y 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 hig her orders has little significance to the manner in which perturbations of the flow can participate in linear and nonlinear resonant interactions. Sin ce we have established that the decay rate of the quasi-modes sharply incre ases with the increase of the wavenumber, one of the major conjectures of t he analysis is that the long-wave components prevail in the large-time asym ptotics of a wide class of initial perturbations, not necessarily the predo minantly 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 qua si-modes also enables us to shed new light on the foundations of the method of piecewise linear approximations widely used in hydrodynamics.