# 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 really property of the fluid in and of itself, and moreover, cannot be a directly measurable physical property of the flow, but should be considered more as a mathematical tool which is derived by taking the curl of the velocity:

The precise meaning is that vorticity is constructed from gradients of the physical flow property velocity, and as such is actually calculated from measured velocity fields, in other words it is indirectly determined.
The same is true for the strain rate and the rotation, both constructs 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.

### Vorticity and the Reynolds Stress Tenzor in Reynolds-Averaged Navier Stokes (RANS)

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.

Rewriting NSE according to 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.
We should notice a few aspects about the appearance of vorticity in that matter. 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 in as much as the RANS formulation and the Reynolds stress definition is concerned, is that even though vorticity is somewhat of a phenomenological aid in understanding the physical picture depicted from the above formulation for the Reynolds stress, it is somewhat problematic to justify the somewhat indirect assumption that Reynolds stresses formulated such are a sufficient phenomenological description of turbulence, as they are merely applied by the imposition of the Reynolds decomposition, meaning averaging NSE after decomposing its flow properties in a form which is known to be phisicaly problematic from the get-go (due to lack of scale separation for example), and beyond that the Reynolds decomposition may be constructed for any stationary time-dependent flow turbulent or not.

It is cumbersome to rationalize an equivalence between time depended chaotic turbulence and a simple velocity correlation that by its nature somewhat lacks the most interesting unsteady features of turbulence.

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

We may formulate the advective form as follows

And a 3-D vorticity formulation of NSE is achieved:

The velocity gradient tensor is often decomposed as a strain rate tensor and a rotation tensor. The extra term that appears in the 3D form (and not in the 2D form) which may be understood as an interaction between vorticity and the velocity gradient tensor is often called the “vortex-stretching term”.

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 that supports the the picture described in the words of Lewis Fry Richardson as

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

Nevertheless, 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.
Is is somewhat a misleading notion to characterize turbulence by eddies breakup, 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. Vorticity is certainly not zero nearly everywhere but it is still important from a phenomenological standpoint to emphasize that constructing our physical understanding or characterization of the turbulence state of the flow solely (as is often done) based on this cartoon of vortical eddies may be ill-advised.

Finally, we should be also careful from an inconsistency that may arise 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?…). in the relations between vorticity and strain rate they are somewhat an artificially contrived contributions to the velocity gradient tensor. Hence, they occur simultaneously and by that their ability of strain rate to “cause” vortex stretching is by no means decisive.

## 2 thoughts on “Vorticity, Vortex Streching and Turbulence”

1. Big whirls have little whirls
That feed on their velocity,
And little whirls have lesser whirls
And so on to viscosity.” [Richardson, 1922]

The inspiring aspect of Richardson’s little poem was his core understanding of the essence of turbulence nearly 100 years ago. The innate understanding of the physics before serious computational or experimental skills is so wonderful. LC

2. The there are a couple of other, less poetic, explanations of turbulence that I quite like too:

“Turbulence is a three-dimensional motion in which vortex stretching causes velocity fluctuations to spread to all wavelengths between a minimum determined by viscous forces and a maximum determined by the boundary conditions of the flow. It is the usual state of fluid motion except at low Reynolds numbers.” [Bradshaw, 1971]

“One can succinctly state that turbulence is a continuous flow phenomenon that is random, rotational, nonlinear, time dependent and three dimensional which occurs in shear flows for real fluids at relatively large mean flow Reynolds numbers to warrant the growth of flow inborn and/or physically induced instabilities and which is further characterized by specific time and space scales determined by the viscous forces and the boundary conditions, a high degree of macroscopic diffusiveness (mixing) and a continuous transfer of energy from larger to smaller spatial scales effected through stretching and tilting of vortex tubes that results, in the end, into turbulence dissipated into heat.” [Sadeh, 1988]

I love that Willy Sadeh says his definition is “succinct”! Cheers, LC