### Motivation:

There are various type of instabilities for a number of flow patterns. The intention of this post is to investigate how these instabilities may trigger laminar-turbulent transition.

I shall do my best to classify these routes leading from a stable laminar to a disturbed one flow, and base my path on the mechanisms responsible for the disturbance growth such that the seemingly random albeit chaotic turbulence flow is apparent.

I shall focus my discussion on three archetypal scenarios in the simplest flow situation of a parallel boundary layer growing in time. Later on I shall try to manifaste examples for more complicated flow patterns and their transition development.

#### Scenario I: Transition due to exponential instabilities:

As explained in Modeling Transition… the traditional method of investigating the stability properties of a particular flow is based on eigenvalue problems.

Transition from an organized laminar flow to a chaotic, seemingly random turbulent state is strongly dependent on a specific non-dimensional number reflecting on how well momentum is diffused relativetothe flow velocity (in the cross-stream direction) and on the thickness of a boundary layer relative to thebody – The *Reynolds Number. *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.

*Orr-Sommerfeld (equation for normal velocity) and Squire equation (for the normal vorticity) for the stability of a parallel undisturbed laminar base-flow*

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.

Concluding the first scenario, Tollmien-Schlichting waves in a Blasius boundary layer or crossflow vortices in three-dimensional boundary layers. It seems then natural that early investigations of transition have used the growth of unstable eigenmodes as the starting point of their analysis. The growth of these exponential primary instabilities, together with subsequent secondary instabilities, in many cases gives a good understanding of how a flow becomes turbulent.

**However,** there are a number of flow situations where difficulties arise. **In some cases, transition to turbulence occurs in a parameter regime** **where no unstable eigenmodes exist**. In other cases, the most unstable exponentially growing mode is observed in experiments.

#### Scenario II: Bypass Transition:

There seem that nowadays we have a much better understanding of disturbance growth and transition scenarios that ** do not** emanate from exponential instabilities. This is best explained in “All About CFD” post: A Forest of Hairpins – on the quest for turbulence coherent structures.

It was Morkovin (in 1969) which announced somewhat invented the expression “bypass transition” as he noted: “Apparently, we can bypass the Tollmien-Schlichting mechanism altogether if it we can find another strongly amplifying mechanism to replace it with…”.

**In my series of four posts**I’ve called such a mechanism Nonmodal instability.

Nonmodal growth mechanisms are associated with the nonnormal structure of the linear stability operator. The basis of the explanation is based on Ellingsen and Palm (1974). Although I shall present the purely temporal view at this point with a strong physical phenomenon orientation, I prefer to base my physical clames onsolid mathematical pilars well known and recognized in stability analyis.

##### The Mathematical Standpoint:

Below is a numerical solution of a The standard technique for explaining the instability of finite difference formulas was developed by von Neumann and others and

described in a 1951 paper of O’Brien, Hyman, Kaplan. ‘Von Neumann analysis’ is another term for discrete Fourier analysis.

One begins by noting that if we ignore the complication of boundary conditions and imagine that the domain is unbounded, then any initial condition for the finite difference formula can be written as a superposition of waves.

For real wave numbers k and corresponding amplification factors (i.e., eigenvalues) λ = λ(k). If |λ| > 1 for some k, we have exponential growth and instability.

It should be very clear though:: Convergence of PDE discretizations depends on norm-boundedness of families of matrices.

It is explainable how von Neumann analysis fits into the general theory of Lax stability:

i.e. the famous Lax Equivalence Theorem, which states that if the discrete approximation is consistent, meaning that it approximates the right PDE as Δx → 0 and Δt → 0, then convergence ⇐⇒ stability.

Here ‘stability’ means that the solution operators are uniformly bounded as the time and space grid sizes approach zero.

A priori, the question of stability requires the analysis of families of matrices, and eigenvalue analysis alone could never give bounds on norms of powers of arbitrary families of matrices. In the special case of constant-coefficient problems on regular grids, however, the Fourier transform takes what would be families of matrices of unbounded dimensions in space into families of matrices of a fixed dimension, indexed over wave numbers. The transformation is unitary, and as a consequence, eigenvalue analysis of the resulting matrices is enough to ensure stability. For practical problems involving boundaries or variable coefficients, further theorems have been proved to show that von Neumann analysis still gives the correct results provided certain additional assumptions are satisfied

such as smoothness of coefficients.

On the other hand, there are some discretizations of PDEs that are fundamentally not translation-invariant. For these, von Neumann analysis is inapplicable, and instabilities may appear that are**nonmodal in nature**.

Let’s have a look at an example:

The first figure reveal that if Δt = O(N −2) as N → ∞, then the maximal norm, though possibly large, is uniformly bounded for all N.

The discretization is Lax-stable, and the numerical solution will converge to the exact solution in the absence of rounding errors. If Δt = O(N −2) as N → ∞, on the other hand, there will be Lax instability

and no convergence. In particular, a choice such as Δt = 0.4N −1 will be catastrophic, even though the eigenvalues in that case remain inside the unit disk for all N. The second figure shows the * pseudospectra* of S for the particular choice N = 20 and Δt = 0.4N −1 = 8N −2 (thus S has dimension 60). Around most of the unit circle, the

**is of modest size, but in the region z ≈ −1 it takes values beyond 106, making it clear that there must be large transient growth. Since the boundary of the pseudospectrum crosses The main question that should be asked is if could one use a discretization of this kind for large-t simulations,**

*resolvent norm***since the instability is transient and dies away eventually? At a glance it might seem so, but the instability can only be expected to be transient for a purely constant-coefficient linear problem in the absence of rounding errors in other words As soon as variable coefficients or nonlinearities or other perturbations are introduced, the loss of convergence is likely to become global.**

Now I shall dive even deeper to ground the mathematics and reconnect directly to the physical phenomenon.

**mathematical conclusions:** Fluid flows that are smooth at low speeds become unstable and then turbulent at higher speeds. This phenomenon has traditionally been investigated by linearizing the equations of flow and looking for unstable eigenvalues of the linearized problem, but the results agree poorly in many cases with experiments. Nevertheless, it has become clear that linear effects play a central role in hydrodynamic instability. A reconciliation of these findings with the traditional analysis can be obtained by considering the **“pseudospectra”** of the linearized problem, which reveal that small perturbations to the smooth flow in the form of streamwise vortices may be amplified by factors on the order of 10 through a **linear** mechanism, **even though all the eigenmodes are stable**. **The same principles apply also to other problems in the mathematical sciences that involve non-orthogonal eigenfunctions**.

##### The Physical Phenomenon Standpoint:

Ellingsen and Palm where the first to show that 3D perturbations may lead to a different kind of instability. One that is independent from an inflection point in the flow.

The process, explained in plan in words (reference shall be remrked), is the usual one. One takes the momentum continuity equations **after linearization** to describe a 3D disturbance which is added to the basic flow. One takes then the divergence of the momentum equatioin and uses continuity to arrive to a Pressure-Poisson equation (PPE from now on… ), and by that we isert it to the equation of the vertical momentum equation (say Y).

In addition we use the definition of the vertical vorticity:

followed by using the momentum equation at directions, say X and Z, and take the derivative of in Z and **respectively**. Now we subtract two get the equations (vertical velocity and vertical vorticity) which describe perfectly the evolution of a3 D disturbance in both time and space.

**The Temporal View**: In the following problem we’ve reached the coordinates Z and X are homogeneous, and the system is linear such that we may work wave numbers and take a good look on special Fourier modes.

To be able to follow that route we take the Fourier transformation of both equations (X and Z), and get the following equations:

From the solution to a properly initialization problem, vertical velocity and the vertical vorticity is achieved and we can find the rest of the velocities by:

When one assumes a non-viscous flow, the equation for the vertical velocity becomes a time dependent Rayleigh equation. For a 2D disturbance and **β**=0 the solution is totally determined by the vertical velocity, the longitudinal X velocity equation can be soved for in assistance of the continuity equation.

Now, if **β** is not zero, well then the disturbance is 3D, and the equation for the vertical vorticity must be taken into account.

In the non-viscous case we may integrate the vertical vorticity equation and get the following result (which we will find very interesting…):

The member who is depicted as I represents the advection of the initial vertical vorticity over the basic field member number II represents the creation of perturbative horizontal velocity by* lift up* of fluid parcels in the presence of basic shear. This one becomes nice and a point to dwell about: for a fourier component α=0 we can calculate the growth explicitly. For this case we may take a look at the vertical velocity, and it’s quite evident that he is not time dependent. The to equations for the vertical velovcity and vertical vorticity becomes:

It follows easily from lookking at the outcome that this the transient **algebraic **growth Ellingsen and Palm where searching for.

## The Lift-Up Effect

This effect is attributed to Landahl, M. T. (1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech.).

There are many ways to explain the effect but I’ve decided to pick the one which is the most straightforward and uses a visualization of the process so naturally.

So let us begin… The most common way to describe this **Lift-up** effect is as a result of fluid parces who hold thier horizontal momentum while pushed by the shear direction. This is what causes horizontal pertubations. Now longitudinal vorticity carries high speed flow to an area of low velocity (say a wall) by which the shear layer is very on ease, and at the same time pushes low speed flow to the high speed area.

some math to better explain:

We shall transform the coordinates so that X1will be in the direction of k=(α,**β**) the second axis Z1will be placed vertically to the wavenumber k.

The new flow will have now two new components on the transformed and new axis given by:

and using :

we may write the following:

Now we may finally give a descriptive presentation of the** Lift-up** effect :

we shall take a small change in w1for a short time δt:

This means that:

Now I find it wonderful. The expression I got for delta(W1) is exactly the horizontal velocity that is induced as a result of the **Lift-up effect** of fluid parcels by the normal velocity, such that the horizontal momentum in a direction perpendicular to the wave-number could be kept.

##### The Lift-up effect presented:

**INITIAL STAGE:**

**FINAL STAGE**:

(Results of this DNS are described in Schlatter and Orlu, 2010, Journal of Fluid Mechanics, 659, 116-126)

#### Scenario III: Oblique transition

In this scenario of transition the

energy is seeded in a pair of oblique waves with Fourier component:

The initial oblique waves are chosen as ** optimal disturbances**.

Moreover, there are random disturbances in nine of the lowest Fourier components. The initial flow consists of a standing wave pattern in the spanwise direction:

As the flow develops, the oblique waves experience transient growth while generating streamwise vortices.

The generated streamwise vortices, in turn, generate streaks by the lift-up effect (see the above precise explanation of the phenomenon: Landahl, M. T. 1980 – A note on an algebraic instability of inviscid parallel shear flows – J. Fluid Mech.) . As the streaks grow, the initial oblique waves start to decay and the flow field is dominated by the streaky structure. From this point on, the development is similar

to the streak breakdown, as a fundamental secondary instability develops on the streak, which starts to oscillate in a sinuous manner, had the original oblique waves not decayed as much as they did, their remaining energy would have triggered a subharmonic varicose streak breakdown (see: A Forest of Hairpins – on the quest for turbulence coherent structures for the mechanism thoroughly explained).

The oblique transition route is best demonstrated by showing more of the second-generation nonlinearly forced Fourier components:

It is appar-

ent that all modes are forced almost equally by the pair of oblique waves.

However, their long term responses are very different. Those marked as (2,2) and (2,0) components and are not very sensitive to forcing, hence grow to a rather low energy. On the other hand, the one marked as the (0,0) component which is a **mean flow modification** does experience quite a **transient** growth. nonetheless, this fourier component is associated with a normal evolution operator and therefore has a low sensitivity to forcing.

Interesting enough, the fourier component marked as (0,2) has a low damping rate (e.g. large transient growth) and is highly sensitive to forcing due to the nonnormal nature of the underlying operator for this wave number combination, as such it becomes the largest response to the forcing by oblique waves.

## Conclusions:

Far and foremost, Transition to turbulence is not a unique process, but one dependent upon the initial or inflow disturbance environment and the type of flow one is interested in.

Landahl’s lift-up effect and its associated generation and breakdown of streaks, on the other hand, are processes that do not correlate strongly with the type of shear flow under consideration. As the initial or otherwise (or both) the inflow condition contain streamwise vortices or disturbances that can generate them by nonlinear means of energy redistribution, large-amplitude streamwise streaks could be expected to be part of the transition process.

One should always keep in mind that the presence of exponential instabilities is highly dependent on the details of the flow under consideration, so each pattern should be exmined individually to determine if exponential instability will overcome the generic growth of streaks. In zero pressure gradient boundary layer (ZPGBL) the growth of streaks and TS-waves are quite comperable in as much as moderate Reynolds numbers are concerned. What shall determine which transition scenario shall prevail are the initial disturbance spectrum, the initial amplitude distribution, and the threshold for secondary instability.

For flow patterns with relatively larger exponential growth rates which with reltively pronounced and quite strong inflection points in the mean velocity profile it is expected that exponential growth to dominate the transition process.