*“Mathematics is the language in which God has written the universe”… Galileo Galilei, Italian astronomer & physicist (1564 – 1642) *

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.

### Natural Transition

Although the laminar flow solution will remain a solution to the Navier-Stokes equations, above a critical Reynolds number it shall become unstable to small disturbances. Then, a series of events shall occur, some of which are linear and some non-linear that act as instabilities in the transitional process and will eventually lead to a fully turbulent state.

The modes of transition onset may be predicted (to some extent) by the linearized stability equations. These equations, derived separately in the beginning of the 20th century by Orr and Sommerfeld (hence termed Orr-Sommerfeld equations) are still investigated by researchers to these day.

Modal analysis performed on the equations may achieve modal solutions characterized by initial exponential growth, such as Klebanoff (K-type, classical), Novosibisrsk or Herbert (N-type or H-type, sub-harmonic) all typical to low intensity of incoming turbulence.

The initial breakthrough in the field of transition onset and process was the description both theoretically and experimentally of what is termed Tollmien-Schlichting wave instabilities for a low incoming turbulence intensity boundary layer.

Under a statistically steady mean flow of low turbelence intensity (tipically <0.5%), transition in an attached boundary layer is initiated by 2D Tollmien-Schlichting instability waves, followed by a 3D instability, followed by the formation of spanwise periodic hairpin vortices.

Then, farther downstream, occurs the breakdown of the laminar layer and the generation of turbulent spots (transition onset). These spots finally coalesce resulting in the formation of a turbulent boundary layer.

It is important to note that this process is a slow process in essence, and moreover, it is very sensitive to all sorts of perturbations which makes the prediction of transition in related applications such as external aerodynamics a delicate matter.

A different growth mechanism characterizing a type of transition onset typical to turbomachinery, where high intensity incoming turbulence exist, is the bypass transition (as to bypassing the Tollmien-Schlichting waves instability, and my favorite transition mechanism 🤓 ).

### Bypass Transition

One of the scenarios for transition process is known as *bypass transition. *This type of transition mechanism is frequently encountered in flows such as those encountered in turbomachinery applications. *
*Under a sufficiently high level of free-stream turbulence intensity, generally >1% , streamwise elongated disturbances are induced in the near-wall zone of an attached laminar boundary layer, termed

**streaks**or

**Klebanoff distortions**. These zones are characterized by high and low velocity, streaky perturbations, alternating in spanwise direction with somewhat of a distinguished periodicity, with a wavelength in the order of the boundary layer thickness.

The streaks are caused by deep penetration of low-frequency disturbances, while the high-frequency disturbances are strongly damped by the laminar shear layer which acts a sort of filter which shear-shelters it from these high-frequency disturbances. The laminar boundary layer distorted by the streaks is susceptible to instabilities, and interestingly, although the streak patterns are of large wavelength, the instability patterns are of short wavelength. This means that the instability patterns can only be excited by high-frequency perturbations, although these are damped by the boundary layer shear. Two of the most noteworthy instability modes are the

*sinuous mode*(much more prevalent, especially in boundary -layer related open flows) and the other

*varicose mode*.

It should be noted that in this stage the boundary-layer is still considered to be laminar.

The Klebanoff distortions grow downstream both in length and amplitude and finally cause breakdown with formation of

**turbulent spots**. This is essentially turbulence onset. These turbulent spots coalesce farther downstream till finally a fully turbulent boundary layer is formed.

Besides the value of free-stream turbulence intensity (again, characterized by Tu intensity > 1%), the streamwise streaks peaks and growth-rate in the streaky zone (still considered laminar) are also affected by the shape of the imposed E(k) spectrum and its integral length scale (i.e same Tu intensity>1% and mean velocity may impose a different transition onset location and spots formation and coalescence).

### Fine grain mechanism description

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.

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.

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.

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.

### Hairpins

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.

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

Some more recommended CFD readings:

## My favorite books list

A must read for turbulence modeling understanding:

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