Nonmodal instability of PDE discretizations – Transient growth through the mathematical standpoint – An introduction primer to bypass transition

Beginnings

The basis of the explanation is based on Ellingsen and Palm (1974), although i have made some alterations for practical reasons.
Rayleigh showed at 1880 that the necessary condition for instability of a non-viscous flow is an Inflection point in the velocity profile. This criterion was sharpened by Fjortof that showed that in addition there has to be also a maxima to the basic vorticity (these to conditions are related to the following: there is a point we call it Ys. At that point u”(Ys)=0 –>Rayleigh . moreover, in the same point Ys Vector(u) in a vector multiplication with the gradient velocity vector<0 –>Fjortof).
All these results refer to a perturbation added to an otherwise stable laminar flow.

3D Flow, and different kinds of instabilities

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.

INITIAL STAGE:

FINAL STAGE

UP NEXT

I hope you ‘d take the time and try to understand as deeply as you can the phenomenology presented, the mathematics you may always return to.

Inthe next Part of our transition voyage, we shall use spectra analysis and learn clearly how transition behaves in basic flows like: Couette, Pipe, and Blausius boundary-layer.