scholarly journals Optimal perturbations and transition energy thresholds in boundary layer shear flows

2020 ◽  
Vol 5 (6) ◽  
Author(s):  
Chris Vavaliaris ◽  
Miguel Beneitez ◽  
Dan S. Henningson
Keyword(s):  
Boundary Layer ◽  
Shear Flows ◽  
Transition Energy ◽  
Energy Thresholds

Collapse transformation for self-focusing solitary waves in boundary-layer type shear flows

1995 ◽  
Vol 206 (3-4) ◽  
pp. 195-202 ◽  
Author(s):  
Dmitry E. Pelinovsky ◽  
Victor I. Shrira
Keyword(s):  
Boundary Layer ◽  
Solitary Waves ◽  
Shear Flows ◽  
Self Focusing

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

2001 ◽  
Vol 446 ◽  
pp. 133-171 ◽  
Author(s):  
VICTOR I. SHRIRA ◽  
IGOR A. SAZONOV
Keyword(s):  
Boundary Layer ◽  
Shear Flows ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Initial Perturbation ◽  
Time Interval ◽  
Two Dimensional ◽  
Leading Order ◽  
Perturbation Field ◽  
Long Wave

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.

The inviscid nonlinear instability of parallel shear flows

1974 ◽  
Vol 63 (4) ◽  
pp. 723-752 ◽  
Author(s):  
J. L. Robinson
Keyword(s):  
Boundary Layer ◽  
Shear Flows ◽  
Matching Condition ◽  
Marginal Values ◽  
Parallel Shear Flows ◽  
Instability Growth ◽  
Velocity Matching ◽  
Viscous Solutions ◽  
Time Dependent Solution ◽  
Qualitative Discussion

In this paper we assume the existence of a nonlinear boundary layer centred on the critical point, and explore its effect on the development of unstable parallel shear flows. A velocity matching condition derived in a qualitative discussion suggests a growth of harmonics which differs from that predicted by previous theories; however, the prediction is in excellent agreement with experimental data. A hyperbolic-tangent velocity profile, subjected to perturbations with wavenumbers and frequencies close to marginal values, is then chosen as a mathematical model of the nonlinear development, both temporal and spatial instability growth being considered.A singularity in the analysis which has been treated in previous theories by the introduction of viscosity is dealt with in the present work by the introduction of a growth boundary layer. The asymptotics are non-uniform and the time-dependent solution does not resemble the steady viscous solutions, even as the growth rate tends to zero. The theory suggests that the instability will develop as a series of temporally growing spiral vortices, a description differing from that of a cat's-eye pattern predicted by existing theories, but in accord with experimental and field observations.

scholarly journals Canonical exact coherent structures embedded in high Reynolds number flows

2014 ◽  
Vol 372 (2020) ◽  
pp. 20130352 ◽  
Author(s):  
K. Deguchi ◽  
P. Hall
Keyword(s):  
Boundary Layer ◽  
Coherent Structures ◽  
Shear Flows ◽  
Wave Interaction ◽  
Vortex Sheet ◽  
Boundary Layer Flows ◽  
Burgers Vortex ◽  
Infinite Region ◽  
High Reynolds ◽  
First Time

The applications and implications of two recently addressed asymptotic descriptions of exact coherent structures in shear flows are discussed. The first type of asymptotic framework to be discussed was introduced in a series of papers by Hall & Smith in the 1990s and was referred to as vortex–wave interaction theory (VWI). New results are given here for the canonical VWI problem in an infinite region; the results confirm and extend the results for the infinite problem inferred the recent VWI computation of plane Couette flow. The results given define for the first time exact coherent structures in unbounded flows. The second type of canonical structure described here is that recently found for asymptomatic suction boundary layer and corresponds to freestream coherent structures (FCS), in boundary layer flows. Here, it is shown that the FCS can also occur in flows such as Burgers vortex sheet. It is concluded that both canonical problems can be locally embedded in general shear flows and thus have widespread applicability.

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