Vorticity and Coherent Structures – Part I

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

Introduction

Although this post shall take great advantage of the notion of vorticity to get a glimpse on the small scale organized fluid elements of significant life-time and scale we call coherent structures.
Now researchers who’s research is to explore the topic tend sometimes to overemphasize vorticity as the physical quantity and derive other physical objects to distinguish organized fluid microelements of significant life-time and scale they term coherent structures.

Vortex structures in a downstream section of a zero-pressure-gradient turbulent boundary
layer(ZPGTBL), computed by means of a DNS. The structures are visualized as iso-surfaces of the second
invariant Q (Q-cretition) of the deformation tensor, where the color indicates the local streamwise velocity.
(Wu and Moin (=2009)

The resulting overemphasizing is very subtle yet extremely important from a the physical phenomenology standpoint. Perhaps quantitatively it seems like it has no meaning whatsoever, but the qualitative nature tells a whole different and quite bizarre story.

The qualitative fallacy stems (sometimes) the initial steps into the world of fluid dynamics.

Each and every engineering student finds himself confronted with the Biot-Savart law at one point or another of their undergraduate studies whether it is related to fluid mechanics or classical electromagnetics. The Biot-Savart carries along the qualitative idea that knowing the curl of a vector field at one point allows us to infer something about the vector field itself at another point. 

Vortex_filament_(Biot-Savart_law_illustration)
The Biot-Savart Law
Biot Savart

As attractive as the idea is, it’s often misleading as it frequently leads to confusion concerning cause and effect. Before even spoken about in Fluid Dynamics 101 it should be pointed out and clearly understood that although vorticity is an extremely important not to say useful object , and has already considered to be a key ingredient of turbulent flows, it is certainly not really property of the fluid in and of itself (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), 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:

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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.

Moreover, the fact that Navier-Stokes equations may be straightforwardly transformed from velocity to vorticity formulation and the use of potential flow related models to create obstructions to the flow strengthens the Biot-Savart frequently inferred view that vorticity induces velocity.

 In the absence of a gravitational or electromagnetic body forces there is no action at a distance in ordinary fluid flows. Casting the equations in one form or another and appealing to the Bio-Savart law as a calculus relation between a vector field and its curl does not mean a vortex at point A can cause a velocity at a remote point B. In conclusion, although the claim that a mathematical relation as the Biot-Savart allows us to infer both quantitative and qualitative information about the velocity field at a distant point is true, in fluid mechanics it does not represent the physics and such a direct cause and effect relation is somewhat misleading as opposed to its counterpart analogy in classical electromagnetics.


Reynolds Stress and Vorticity – Motivation

After clearing the table from inconveniences we may now shift our attention the main issue of the post without any lack of confidence…
As pointed in the former section vorticity is formally defined as the curl of the velocity field:

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And under the conceptual phenomenological restrictions raised in the former chapter vorticity may be interpreted as a measure of the rotation of a fluid element. keeping these restrictions in mind, and for ease we usually characterize turbulence as “chaotic vorticity”. To get a fifirst notion on the importance of vorticity we rewrite the Reynolds stress as:

The second term on the right is a gradient of the scalar:

and when the complete equations of motion for the averaged ui is considered, this term can be added to the pressure gradient.
The first term on the right is quite interesting, and it will be shown to express the fact that the Reynolds stress implies the existence of fluctuating vorticity.

With or without any imposition of restrictions on the qualitative phenomenological nature of vorticity, a flow without a zero curl of the physical property velocity (e.g. vorticity) could not be considered turbulent.

Vorticity and the Reynolds Stress Tensor 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

2

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

3

and substitute it to the former equation

4

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

5

Rewriting NSE according to the decomposition delivers:

6.png

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 physically 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.


How About Vortex stretching

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).

vortex stretching

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

7

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

8

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

9.png

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

10

We may formulate the advective form as follows

11

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

12

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”…

richardson


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.


The Return of The Vorticity Equation

The equation for vorticity can be derived from the Navier–Stokes equations by applying the curl to both sides of the equation. Clearly, the pressure term vanishes because the curl of a gradient is identical zero. The result reads:

Vorticity equation

As explained in the introduction the equation assists in the understanding of how the vorticity changes as we move along with a fluid element.
The second term on the right-hand side is recognized as the diffusion of vorticity due to viscosity.
The first term on the right is a fundamental quantity derived from velocity gradients such that it represents the strain rates (a somewhat different manipulation of the velocity gradients shall derive the fundamental quantity of rotation). We get to this term by rewriting the first term in the right hand side of the vorticity equation as such:

As:

Is the quantity derived from velocity gradients such that it represents the strain rates, and in as much as we’ve chosen vorticity as the representative object it describes the interaction between the fluid deformation and this representative object.

The term in the right hand side of the vorticity equation is the material derivative of the vorticity is identical to the equation for a material line segment δXi in a flow:

hence it somewhat straight forward to think of it as the motion of such a line segment which changes by orientation and length under the action of the flow (another analogy mind you…). This is analogous to the evolution of the vorticity wi.
Under this simply understood analogy (line segment vs. vorticity) two types of interactions could be distinguished:

  • i equal j:

    then the rate of strain describes the change in length of a material line segment. if the strain rate is grater than zero (sij>0) the line segment is stretched, otherwise, if the strain rate is less than zero (sij<0) the line segment is compressed.

    for example, if we take a situation where a vortex line: wj=(0,0,w) with a velocity field which yields w33 is nonzero and viscosity is negligible.
    The vorticity equation reduces to:

With a solution to differential equation:

With such conditions for the flow pattern and the derived particular solution vorticity increases or decreases depending on the sign of s33.
When s33 >0 the vorticity increases. This is what turbulence researchers refer to as Vortex Stretching.

Some would claim that next to the increase of vorticity, it appears that an energy transfer takes place from the deformation fifield to the vortex line. In other words, by vortex stretching, the vortex spins up. The energy needed for this is evidently supplied by the deformation field.

Vortex Stretching (as per example)

A word about the claim of the former paragraph is in order.

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:

7

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

8

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.

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

9.png

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:

10

We may formulate the advective form as follows:

11

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

12

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 what we’ve coined in this post and former example as the “vortex-stretching term”.

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.

It 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.



still, vorticity, and in particular vortex stretching, is a fundamental process in turbulence. It is clear how vorticity plays an important part in the stability of flows (see the three scenarios, especially the third scenario in “Transition to Turbulence Series – Transition Scenarios” or a violent Rayleigh-Be’nard convection turbulence scenario).
Due to these instability processes the larger eddies are created, determining the macrostructure. In turn, these larger eddies supply energy to the smaller eddies through the cascade process (the mechanism is evident albeit not precise as explained before).

In conclusion of the first type of interaction (beside some remarks that should be taken into account from a physical qualitative perspective certainly, but also have a quantitative effect), The larger eddies deform the smaller eddies, and so the vorticity of the smaller eddies increases, while at the same time energy from the larger eddies is transferred to the smaller eddies. An important consequence of this is that we encounter the largest vorticity magnitude at the microstructure, these organized fluid microelements of significant life-time and scale termed coherent structures.

It’s important to raise the point of initially starting with 2D only for convenient and ease understanding gradually but turning back to the 3D representation was essential since the process of vortex stretching can only occur in three-dimensional flows. In two-dimensional flows vortex lines cannot be deformed. Turbulence as described here can not exist in two dimensions. A very good example we find in the wave motion at the surface. It is not turbulent.

The flow behind a grid in a thin soap film (nearly 2D) – there is no effective energy cascade even though some vortex pairing are visible
(M.A. Rutgers – 1998)

Moreover, Since vorticity is a vector quantity (curl of a the velocity field by defiinition), it is not such that lends itself to graphical visualization. In the case two-dimensional flow, the vorticity component normal to the plane of observation is plotted as a scalar quantity. In the case of a three-dimensional representation, vortical structures are visualized as I’ve shown in the famous Wu-Moin ZPGBL by using the Q-criterion defined as the second invariant of the deformation tensor:

It a valuable representation as for the first invariant P=0 in incompressible flow, such that:

The immediate consequence is that vortical regions are characterized by large values of Q. There must be added another criterion for the pressure to distinguish between eddies and shear layers.

There is also a third determinant R of the deformation tensor which is directly given by its determinant and is essentially the product of the deformation tensor eigenvalues:

R=-det(dui/dxi)

A plot of the QR probability density function shows a characteristic shape, and a plot of a Q-R diagram should be validated:

logarithmic contour intervals of Q-R probability density function (normalized with respect to the mean rotation vector – also derived from manipulating the gradients of the velocity)
I: stable stretching
II: unstable compressing
III: unstable
node/saddle/saddle
IV: stable node/saddle/saddle
(Ooi et al. – 1999)

Coherent Structures

Admittingly, despite my former caution remarks, vortex stretching is certainly very useful to get a first impression of how turbulent eddies develop.
If we consider a regular near wall flow (u(y),0,0), the velocity measurements close to the wall will show regions where the velocity is small in comparison to the surroundings.These are called low-speed streaks and these are characteristic for all near-wall turbulent flows (again, see the complete development of thescenario in “Transition to Turbulence Series – Transition Scenarios“)

(a) Development of low speed streak becoming perturbed
(b) perturbation grows due to Kelvin-Helmholz instability
(c) Vortex sheet around a low spead streak rolls into a hairpin
(d) The mean deformation keeps stretching the hairpin vortices until bursting to a fully unstable state.

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