Linear stability of shear profiles and relation to the secondary instability of the Dean flow

Cunff, Cédric Le; Bottaro, Alessandro

doi:10.1063/1.858555

Cited by 14 publications

(7 citation statements)

References 17 publications

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“…As observed in a number of experiments and numerical studies, see Swearingen & Blackwelder (1987) and LeCunff & Bottaro (1993) for example, the sinuous instability can be related to the spanwise inflectional points of the mean flow. Andersson et al (2001) have shown that the streamwise velocity of the secondary instability modes is concentrated around the critical layer, i.e.…”

Section: Eigenmode Structure and Streamwise Vorticity Generationmentioning

confidence: 78%

Transition of streamwise streaks in zero-pressure-gradient boundary layers

2002

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A transition scenario initiated by streamwise low- and high-speed streaks in a flat-plate boundary layer is studied. In many shear flows, the perturbations that show the highest potential for transient energy amplification consist of streamwise-aligned vortices. Due to the lift-up mechanism these optimal disturbances lead to elongated streamwise streaks downstream, with significant spanwise modulation. In a previous investigation (Andersson et al. 2001), the stability of these streaks in a zero-pressure-gradient boundary layer was studied by means of Floquet theory and numerical simulations. The sinuous instability mode was found to be the most dangerous disturbance. We present here the first simulation of the breakdown to turbulence originating from the sinuous instability of streamwise streaks. The main structures observed during the transition process consist of elongated quasi-streamwise vortices located on the flanks of the low-speed streak. Vortices of alternating sign are overlapping in the streamwise direction in a staggered pattern. The present scenario is compared with transition initiated by Tollmien–Schlichting waves and their secondary instability and by-pass transition initiated by a pair of oblique waves. The relevance of this scenario to transition induced by free-stream turbulence is also discussed.

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Section: Eigenmode Structure and Streamwise Vorticity Generationmentioning

confidence: 78%

Transition of streamwise streaks in zero-pressure-gradient boundary layers

2002

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“…The sinuous streak instability is induced by the mean spanwise shear on the sides of the region of defect velocity as in classical two-dimensional wakes past a bluff body. It is therefore instructive to compare the instability properties of the streaky base flows with those of two-dimensional wakes (LeCunff & Bottaro 1993). At this point, we may note that for the bluff body, the wake and the resulting instability are isolated in the spanwise direction, whereas the streak instability arises on a spanwise periodic basic flow.…”

Section: Discussionmentioning

confidence: 99%

On the convectively unstable nature of optimal streaks in boundary layers

et al. 2003

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International audienceThe objective of the study is to determine the absolute/convective nature of the secondary instability experienced by finite-amplitude streaks in the flat-plate boundary layer. A family of parallel streaky base flows is defined by extracting velocity profiles from direct numerical simulations of nonlinearly saturated optimal streaks. The computed impulse response of the streaky base flows is then determined as a function of streak amplitude and streamwise station. Both the temporal and spatio-temporal instability properties are directly retrieved from the impulse response wave packet, without solving the dispersion relation or applying the pinching point criterion in the complex wavenumber plane. The instability of optimal streaks is found to be unambiguously convective for all streak amplitudes and streamwise stations. It is more convective than the Blasius boundary layer in the absence of streaks; the trailing edge-velocity of a Tollmien-Schlichting wave packet in the Blasius boundary layer is around 35% of the free-stream velocity, while that of the wave packet riding on the streaky base flow is around 70%. This is because the streak instability is primarily induced by the spanwise shear and the associated Reynolds stress production term is located further away from the wall, in a larger velocity region, than for the Tollmien-Schlichting instability. The streak impulse response consists of the sinuous mode of instability triggered by the spanwise wake-like profile, as confirmed by comparing the numerical results with the absolute/convective instability properties of the family of two-dimensional wakes introduced by Monkewitz (1988). The convective nature of the secondary streak instability implies that the type of bypass transition studied here involves streaks that behave as amplifiers of external noise

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“…In addition there is a second instability, called undulating, that occurs if R > 1.3R c . By considering one-dimensional streak profiles of the form U(y) and U(z), Le Cunff & Bottaro (1993) concluded that the undulating instability is centrifugal in nature. Matsson & Alfredsson (1992) observed the spanwise inflectional instability experimentally.…”

Section: Streak Breakdownmentioning

confidence: 99%

On stability of streamwise streaks and transition thresholds in plane channel flows

et al. 1998

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Streak breakdown caused by a spanwise inflectional instability is one phase of the following transition scenarios, which occur in plane Poiseuille and Couette flow. The streamwise vortex scenario is described by (SV) streamwise vortices =⇒ streamwise streaks =⇒ streak breakdown =⇒ transition.The oblique wave scenario is described by (OW) oblique waves =⇒ streamwise vortices =⇒ streamwise streaks =⇒ streak breakdown =⇒ transition.The purpose of this paper is to investigate the streak breakdown phase of the above scenarios by a linear stability analysis and compare threshold energies for transition for the above scenarios with those for transition initiated by Tollmien-Schlichting waves (TS), two-dimensional optimals (2DOPT), and random noise (N) at subcritical Reynolds numbers. We find that if instability occurs, it is confined to disturbances with streamwise wavenumbers α 0 satisfying 0 < α min < |α 0 | < α max . In these unstable cases, the least stable mode is located near the centre of the channel with a phase velocity approximately equal to the centreline velocity of the mean flow. Growth rates for instability increase with streak amplitude. For Couette flow streak breakdown is inhibited by mean shear. Using the linear stability analysis, we determine lower bounds on threshold amplitude for transition for scenario (SV) that are consistent with thresholds determined by direct numerical simulations.In the final part of the paper we show that the threshold energies for transition in Poiseuille flow at subcritical Reynolds numbers for scenarios (SV) and (OW) are two orders of magnitude lower than the threshold for transition initiated by Tollmien-Schlichting waves (TS) and an order of magnitude lower than that for (2DOPT). Scenarios (SV) and (OW) occur on a viscous time scale. However, even when transition times are taken into account, the threshold energy required for transition at a given time for (SV) and (OW) is lower than that for the (TS) and (2DOPT) scenarios at Reynolds number 1500.

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Linear stability of shear profiles and relation to the secondary instability of the Dean flow

Cited by 14 publications

References 17 publications

Transition of streamwise streaks in zero-pressure-gradient boundary layers

Transition of streamwise streaks in zero-pressure-gradient boundary layers

On the convectively unstable nature of optimal streaks in boundary layers

On stability of streamwise streaks and transition thresholds in plane channel flows

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