# Vorticity, Vortex Streching and Turbulence

“Big whorls have little whorls that feed on their velocity, and little whorls have lesser whorls and so on to viscosity” – Lewis Fry Richardson

After reading Is velocity induction by vorticity a fallacy?, we clearly understood that although vorticity is an extremely important topic, and has already considered to be a key ingredient of turbulent flow, it is certainly not a property of the fluid and moreover, cannot be directly measurable physical property of the flow. It’s merely a mathematical endeavour which is the curl of the velocity

The precise meaning is that mathematical vorticity is built from gradients of the physical velocity of the physical flow property velocity. The gradients themself are a contract from measure velocity fields, meaning an indirect determination of vorticity.
The same is true for the strain rate and the rotation, both a construct of the velocity gradient tensor. i.e., a set of fundamental quantities is a set of velocity gradients such that vorticity and strain rates are derived quantities.

### The Manipulation of casting the equations in different forms

To demonstrate the connection between the Reynolds stresses and turbulence in the context of RANS, a good start is to write the equations in tensor notation

Express the advective term (in the left hand side) by manipulating it to

and substitute it to the former equation

Now introducing the Reynolds decomposition according to which a flow variable is decomposed into mean and fluctuating quantities.

And using the former NSE with the decomposition delivers

This seems like an expression of the fluctuating quantities related to the Reynolds stresses in terms of mean quantities including the vorticity, as to emphasize the importance of the vorticity in the generation of Reynolds stresses.

And yet again our handling of vorticity presents some basic flaws. First introduction of vorticity to NSE is purely artificial as it by no way reflect actual physics (how could it be it was obtained by adding zero to the momentum equation).
The second, much more obvious, is that even if one could argue vorticity could stand on a solid physical grounds there is no way to justify the somewhat indirect assumption that Reynolds stresses are turbulence, as they are merely applied by the imposition of the Reynolds decomposition, meaning averaging NSE, simply a mathematical endeavour, beyond that the Reynolds decomposition may be constructed for any stationary  time-dependent flow turbulent or not.

Summing that up, stresses are time averaged quantities hence independent on time, nonetheless that only the normal stresses appear in the formulation.
The final argument is that it is cumbersome to rationalize an equivalence between time depended chaotic turbulence and a simple velocity correlation that by its nature is time independent.

A very known mechanism for the transfer of energy between small wave (large energetic scales) numbers to high wave numbers (small scales) by is done by vortex stretching.
A vortex tube subjected to strain from local velocity gradients of the flowfield will tend to stretch, thereby shrinking its diameter. The consequence is that the energy associated with that vortex is acting at a larger wave number (smaller scales).

The easiest way to test this phenomenon is to work through the 2-D vorticity equation and identify mechanisms that could generate such behaviour.

I shall consider first the 2-D vorticity equation. I shall begin with the 2D NSE and compute the curl

It is easy to observe that in the 2-D case only one component of the vorticity is non-zero and hence a scalar.
Remembering the strain rate tensor

From these we may clearly see that turbulence can not be 2-D. There is then no mechanism to endure vortex stretching in the 2-D case of the vorticity transport equation.

Going to the 3-D case, I shall again take the curl of NSE but now result with a vector

The first term on the right hand side is identically zero since it’s a curl of a gradient. We make a bold move (assuming smoothness) to allow us commutation between the curl and the time derivative and handle the following

Now taking a scrutiny look on the advective terms

Now I the 3-D vorticity equation, my casting of NSE in a different form may be presented

The velocity gradient tensor is often decomposed as a strain rate tensor and a rotation tensor. The term that is associated with the vorticity and the strain rate tensor is often called the “vorticity stretching term”.

The misleading notion was already mentioned in Is velocity induction by vorticity a fallacy?. It is very often for many to emphasize vorticity as actual vortical structures of what most known as “eddies” to explain the nature of turbulence.
In particular we have achieved a 3-D vorticity equation of which vortex stretching which supports this energy cascade as in the words of Lewis Fry Richardson:

“Big whorls have little whorls that feed on their velocity, and little whorls have lesser whorls and so on to viscosity”…

Thereby casting the equation the 3-D vorticity equation that supports such a cascade when it is now known that the simple idea of vortex stretching and subsequent breaking into yet smaller vortices (of which we call “eddies”) is not an accurate picture of actual physics. Moreover, the example of wall-bounded shear flows has shown that as much as part of the energy does cascade to small scales nearly a third is “back scattered” up to the large scales.
But perhaps the most misleading notion comes from the cartoon characterization of turbulence by eddies, as in a given instant of the vast large of turbulent flows only a fraction of the volume is occupied by such creatures, hardly rendering such a picture of turbulent flow by the description of these eddies as a reliable representation of turbulence flow.
The last does not imply that the vorticity is zero nearly everywhere but
rather serves to emphasize that constructing a physical theory based on this cartoon of vortical eddies may be ill-advised.

Now to finalize, we observe the inconsistency in the attempt to view strain rate as the cause of vortex stretching. NS equations are filled with circular cause and effect reciprocal relations (does a pressure field cause a velocity field or is it the other way around?…)
For the case of manifesting the reciprocity relations between vorticity and strain rate they are simply artificially contrived contributions to the velocity gradient tensor. Hence, they occur simultaneously and by that their ability to strain rate to “cause” vortex stretching is by no means decisive.

## This blogger favorite book list

A must read for turbulence modeling understanding:

(Including with F. Menter references)