Floquet analysis of secondary instability of boundary layers distorted by Klebanoff streaks and Tollmien–Schlichting waves

Three-dimensional (3D) linear stability properties are considered for steady and unsteady 2D or 3D boundary layers with significant non-parallelism present. Two main examples of such non-parallel flows whose stability is of interest are, firstly, steady motion, over roughness elements, in cross flow, or in large-scale separation and, secondly, unsteady 2D Tollmien-Schlichting (TS) motion, with its associated question of secondary instabilities. A high-frequency stability analysis is presented here. It is found that, for 2DTS or steady boundary layers, there is a swing in the direction of maximum TS spatial growth rate, from 0° for parallel flow towards 64.68° away from the free-stream direction, as the nonparallel flow effects increase. These effects then depend principally on, and indeed are proportional to, the local slope of the boundary-layer displacement. Cross flow can also have a profound impact on TS instabilities. Further implications for higher-amplitude and/or fasterscale disturbances, their secondary instability, and nonlinear interactions, are also discussed.

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Secondary instability of cross-flow vortices in Falkner–Skan–Cooke boundary layers

Journal of Fluid Mechanics ◽

10.1017/s0022112098001931 ◽

1998 ◽

Vol 368 ◽

pp. 339-357 ◽

Cited By ~ 63

Author(s):

MARKUS HÖGBERG ◽

DAN HENNINGSON

Keyword(s):

Growth Rate ◽

Shear Layer ◽

Boundary Layers ◽

Growth Rates ◽

High Frequency ◽

Cross Flow ◽

Low Frequency ◽

Secondary Instability ◽

The Cross ◽

Flow Vortex

Linear eigenvalue calculations and spatial direct numerical simulations (DNS) of disturbance growth in Falkner–Skan–Cooke (FSC) boundary layers have been performed. The growth rates of the small-amplitude disturbances obtained from the DNS calculations show differences compared to linear local theory, i.e. non-parallel effects are present. With higher amplitude initial disturbances in the DNS calculations, saturated cross-flow vortices are obtained. In these vortices strong shear layers appear. When a small random disturbance is added to a saturated cross-flow vortex, a low-frequency mode is found located at the bottom shear layer of the cross-flow vortex and a high-frequency secondary instability is found at the upper shear layer of the cross-flow vortex. The growth rates of the secondary instabilities are found from detailed analysis of simulations of single-frequency disturbances. The low-frequency disturbance is amplified throughout the domain, but with a lower growth rate than the high-frequency disturbance, which is amplified only once the cross-flow vortices have started to saturate. The high-frequency disturbance has a growth rate that is considerably higher than the growth rates for the primary instabilities, and it is conjectured that the onset of the high-frequency instability is well correlated with the start of transition.

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A numerical evaluation of the asymptotic theory of receptivity for subsonic compressible boundary layers

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.196 ◽

2015 ◽

Vol 771 ◽

pp. 520-546 ◽

Cited By ~ 7

Author(s):

Nicola De Tullio ◽

Anatoly I. Ruban

Keyword(s):

Mach Number ◽

Boundary Layers ◽

Asymptotic Theory ◽

Stokes Equations ◽

Roughness Element ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Compressible Boundary Layers ◽

Tollmien Schlichting ◽

Low Amplitude

The capabilities of the triple-deck theory of receptivity for subsonic compressible boundary layers have been thoroughly investigated through comparisons with numerical simulations of the compressible Navier–Stokes equations. The analysis focused on the two Tollmien–Schlichting wave linear receptivity problems arising due to the interaction between a low-amplitude acoustic wave and a small isolated roughness element, and the low-amplitude time-periodic vibrations of a ribbon placed on the wall of a flat plate. A parametric study was carried out to look at the effects of roughness element and vibrating ribbon longitudinal dimensions, Reynolds number, Mach number and Tollmien–Schlichting wave frequency. The flat plate is considered isothermal, with a temperature equal to the laminar adiabatic-wall temperature. Numerical simulations of the full and the linearised compressible Navier–Stokes equations have been carried out using high-order finite differences to obtain, respectively, the steady basic flows and the unsteady disturbance fields for the different flow configurations analysed. The results show that the asymptotic theory and the Navier–Stokes simulations are in good agreement. The initial Tollmien–Schlichting wave amplitudes and, in particular, the trends indicated by the theory across the whole parameter space are in excellent agreement with the numerical results. An important finding of the present study is that the behaviour of the theoretical solutions obtained for $\mathit{Re}\rightarrow \infty$ holds at finite Reynolds numbers and the only conditions needed for the theoretical predictions to be accurate are that the receptivity process be linear and the free-stream Mach number be subsonic.

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The production of subharmonic waves in the nonlinear evolution of wavepackets in boundary layers

Journal of Fluid Mechanics ◽

10.1017/s0022112099006424 ◽

1999 ◽

Vol 399 ◽

pp. 301-318 ◽

Cited By ~ 37

Author(s):

MARCELLO A. F. MEDEIROS ◽

MICHAEL GASTER

Keyword(s):

Boundary Layer ◽

Boundary Layers ◽

Nonlinear Interaction ◽

Strong Influence ◽

Nonlinear Evolution ◽

Nonlinear Behaviour ◽

Two Dimensional ◽

Tollmien Schlichting ◽

Acoustic Excitations ◽

Dimensional Wave

The nonlinear evolution of wavepackets in a laminar boundary layer has been studied experimentally. The packets were generated by acoustic excitations injected into the boundary layer through a small hole in the plate. Various packets with different phases relative to the envelope were studied. It was found that for all the packets the nonlinearity involved the appearance of oblique modes of frequency close to the subharmonic of the dominant two-dimensional wave. Moreover, the results confirmed that the phase had a strong influence on the strength of the nonlinear interaction. The experimental observations also indicated that although a subharmonic resonance appeared to be present in the process, it alone could not explain the nonlinear behaviour. The experiment demonstrated that the process must also involve a mechanism that generates oblique waves of frequency lower than the Tollmien–Schlichting band.

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Experiments in a boundary layer subjected to free stream turbulence. Part 2. The role of TS-waves in the transition process

Journal of Fluid Mechanics ◽

10.1017/s0022112094003095 ◽

1994 ◽

Vol 281 ◽

pp. 219-245 ◽

Cited By ~ 80

Author(s):

A. V. Boiko ◽

K. J. A. Westin ◽

B. G. B. Klingmann ◽

V. V. Kozlov ◽

P. H. Alfredsson

Keyword(s):

Boundary Layer ◽

Boundary Layers ◽

Small Amplitude ◽

Free Stream ◽

Transition Process ◽

Natural Occurrence ◽

Free Stream Turbulence ◽

Onset Of Turbulence ◽

Tollmien Schlichting

The natural occurrence of Tollmien-Schlichting (TS) waves has so far only been observed in boundary layers subjected to moderate levels of free stream turbulence (Tu < 1%), owing to the difficulty in detecting small-amplitude waves in highly perturbed boundary layers. By introducing controlled oscillations with a vibrating ribbon, it is possible to study small-amplitude waves using phase-selective filtering techniques. In the present work, the effect of TS-waves on the transition is studied at Tu = 1.5%. It is demonstrated that TS-waves can exist and develop in a similar way as in an undisturbed boundary layer. It is also found that TS-waves with quite small amplitudes are involved in nonlinear interactions which lead to a regeneration of TS-waves in the whole unstable frequency band. This results in a significant increase in the number of turbulent spots, which promote the onset of turbulence.

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