Nonmodal instability of PDE discretizations – Spectra Analysis – An Introduction Primer for ByPass Transition

In this section we’re going to pull out a few tools from our analysis arsenal and pay very close attention to what might seem at first like odd results. Doing that would be best on well known basic flows, i.e. Plane Poiseuille, and then Blasius boundary layer, pipe flow and coutte.

Plane Poiseuille

It is possible to numerically find the eigenvalues (spectral colocation), to find three branches on the imaginary C plane: A(Cr–>1), P(Cr–>1), S(Cr about 2\3).

The thing is that there are eigenvalues which penetrate the upper left plane and point to a modal instability, only that the parabolic velocity profile of Plane Poiseuille flow has no inflection point.

This means to according to Rayleigh necessary criterion for Reeynolds—>∞ we conclude that the vorticity is acting as a destabilizing feature (which is contradictory to our intuition).
This unstable mode is the “Tollmien Schlichting Mode” (Orszag >5772).
for α=0 it may be seen that only one branch is apparent in the lower imaginary plane. This case allows us to get a dispersion ratio analytically and for Orr-Sommerfeld modes:

while µ fulfils:

Coutte, Pipe, Balusius

The pipe has actually the same branches as the plane poiseuille, the difference is tha Couette has two A branches with no P branch (see above). For every A branch there is always another “twin”. Both spectrums are stable for all Reynolds numbers in the modal sense.

In Blausius BL we obtain numerically what looks like a reunion between the S and P branches. Numerically it is only a discretized representation to the continuous spectrum that we get when easing the boundary conditions (actually a bound to in ∞ instead of damping).
The A branch has only one unstable mode at most (depending on the chosen parameters) and this is a Tollmien-Schlichting mode (maximum amplitude and high velocity near the wall).

Stability Character of a Continuous Stability Operator

for plane Poiseuille flow we get a discrete set of eigenvalues for each set of parameters (α, β, Re). Changing α and β changes the complex frequencies ω change accordingly but still always occupy the complex plain with great density.

For pipe flow there accepts no dense area, just a bunch of close by lines. This is because the azimuth wave-number n accepts only natural numbers.

PART II: Conlusion

Specta analysis is a basic and mandatory tool to relate phenomenon that many times could not just simply jump out of Orr-Sommerfeld/Squire equation.

I know this is quite a hard post for people unfamiliar, this is why I’ve decided to not overload and balance with the complexity and its significance what shall come next:

PART III: Transient Growth, The viscous counterpart, and localized disturbances.

Part IV: will take a different point of view while presenting Floquet analysis.

And then… who knows… 🌈

The best thing that each of you may achieve is to see the strings attaching one by one into one of the most beautiful and intriguing fields in fluid dynamics.

Leave a Reply