Nonmodal instability of PDE discretizations  – The Need For Linear Growth Mechanisms – Prolog

Turbulence is the name given to imperfectly understood class of chaotic solutions to the NSE in which many (and denied a spectral gap) degrees of freedom are excited (If we call a dog’s tail a leg than how many legs does a dog have?… It’s still four!)

H. Aref 1999.

This post serves as somewhat of a prolog to what’s to come. it allows us to understand
the understanding of the motivation for aquiring the mathematical toolkit and the physical path otherwise undermined to ever have a chance of onhoding it at the palm of our hands.

The Reynolds-Orr equation

The Reynolds-orr was based on a critical criterion found, and actually introduced basing on disturbance energy. At first glance it seems very intuitive do derive an evolution equation for the kinetic disturbance, and indeed it is.

After a few ordinary mathematical procedures which I shall not repeat here (see Applied Mathematical Sciences volume 142):

All the terms that could be written as gradients vanish when integrated over the volume V. In particular it is important to realize that nonlinear terms have dropped out. due a multiplication ui linear terms resulted in quadratic terms, while a nonlinear term after multiplication corresponds to a cubic term. The two remaining terms of the equation above represents exchange of energy with the base flow and of course energy dissipation due to viscous effects.

The Cruciality of The Linear Growth Mechanism

The Reynolds-Orr equation can carry us to an important conclusion about the behavior of various growth mechanisms for disturbances superimposed on base flows that are solutions of the Navier-Stokes equations.

We can clearly see that in part of the derivation (After a few ordinary mathematical procedures which I shall not repeat here (see Applied Mathematical Sciences volume 142) the instantaneous growth rate:

The first obvious impression to remember for it’s still an undergoing debate. In other words, what we come to observe is that the above instenious growth is independent of the disturbance amplitude. Simply put, the growth rate of a finite amplitude disturbance can at each instant of its evolution may be found from an infinitesimal disturbance with the same shape.

Now let me mark this down… The instantaneous growth rate of a finite amplitude disturbance are given by mechanisms that are linearized equations, and the total growth a final amplitude disturbance can be regarded as a summation of grows rates associated with linear mechanisms. This is actually a direct consequence of the conservative nature of the nonlinear terms in Navier-Stokes equations.

A Word relating To Linear versus nonlinear processes

We should pay close attention to the fact that all the arguments made above assume a linear energy source. Moreover, it assumes that the total disturbances is superimposed on a laminar base flow. Well, that satisfies the Navier-Stokes equations.
On the other hand, it is possible and very common to measure disturbances on a mean flow already modified by these disturbances.

No matter the base flow at hand (either modified or still non-modified by the disturbance) the only mechanism to extract energy for continuous growth (albeit transient) of a final sized disturbance is the linear naechnism while nonlinear mechanism perform a role of redstributing the energy between different directions.