A Forest of Hairpins – on the quest for turbulence coherent structures

Recent studies, suggest a dominant role for hairpin shaped vortices of various sizes throughout the boundary layer in transitional stages. Hairpins are well-known structures in transitional flows, appearing as a result of shear-layer roll-up.
One of the scenarios for transition process is known as bypass transition. This mechanism of transition is important for boundary layer flows of which high levels of turbulence in the free stream occur and dominated by diffusion effects as turbulence is diffused into the boundary layer from high free stream levels. Such a transition mechanism is often encountered in turbomachinery applications for example.

Turbulent Boundary Layer (P. Schlatter and D. Henningson of KTH)

 The imposed mechanism could be described by an instability of a finite amplitude “saturated” state added to an otherwise laminar stable base flow, its modal stability described by well-known Orr-Sommerfeld/Squire (OS/SQ) equations. While the disturbance consisting of separately damped Orr-Sommerfeld/Squire modes (asymptotically stable in time) they seem to grow transiently as a superposition, by-product of highly non-normal (large condition number) OS operator.

SSP

Sustained turbulent coherent structures (stream-wise streaks)  –
regeneration cycle of near wall turbulence

A view of the process may be found through Landahl’s “Lift-Up effect” in a nominally zero-pressure gradient boundary layer subject to high levels of free stream turbulence of which the mechanism of Tollmien-Schlichting waves transition is bypassed as presented in the figure below.

Landahl TG

Bypass transition mechanism description

The finite (albeit small) disturbance energy growth would be achieved by a linear mechanism, the only mechanism available for energy growth at this stage, while the non-linearity merely redistributes this
energy between the different scales.

Transient Growth

Linear transient growth outline

The disturbance growth continues until “saturation”, establishing a base flow distortion as a new “quasi-steady state” (quasi as long as the next instability has a much faster growth rate then the characteristic time of the quasi-state) with longitudinal velocity streaks (“Pseudo-modes” of Orr-Sommerfeld/Squire Eq. or otherwise modes of a distorted operator but in a way which creates only a base flow variation, still non-normal by himself). The flow stands yet another form of “Linear”, inviscid instability (“Varicose” – wall normal) for hairpins to be generated.

The role of the transient growth mechanism in the onset of transition of this kind is a essentially a question of the spatiotemporal reciprocal relations between it and the nonlinear redistribution of energy of mode interaction which is reached as the transition process advances.

zpgfpbl

DNS of a zero-pressure-gradient flat plate boundary layer (P. Moin and X. Wu)

The hypothesis that these hairpin shaped vortex structures are an inseparable part at transitional stages (e.g. “Forest of Hairpins”) is confirmed by DNS of a zero-pressure-gradient flat plate boundary layer (P. Moin and X. Wu). Moreover, they seem to persist (though less dominating the Flow as Reynolds number is increased) into the turbulence stage.

Turbulent Boundary Layer (APS Gallery Submission)

For in depth understanding of the transition concept I would strongly recommend:

The end


A book extraordinary recommendation from a CFD blogger:


Some more recommended CFD readings:

This blogger favorite book list

A must read for turbulence modeling understanding:

(Including with F. Menter references)

LOGOTENZOR – Channel Partner ANSYS

TENZOR (authorised channel partner ANSYS) – Turbulence modeling courses

TAZTAZ – design, analysis and optimization to the level of art

 

 

7 thoughts on “A Forest of Hairpins – on the quest for turbulence coherent structures

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