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(ε−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
O(ε4) (O(ε4) 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.
Particle movement in a boundary layer
