Why does a turbulent flow form eddies?

I suggest to think about two elements a) the vorticity w equation b) the vortical tube (the envelope of the vorticity vector field).

You can see that a vortical tube has the properties of having

Int n.w dS =constant

therefore the intensity of the vorticity increases when the section of the vortical tube reduces. In turbulence, the vorticity is a 3D field and exists the stretching term that acts to change both direction of the vortical tube and its intensity. This way, large vortical structure at the integral scale will be fragmented in smaller ans smaller vortical structures. This is the genesis of the inertial cascade, that is the presence in the flow of eddies of differente sizes. Dissipation of such eddies appears only at small scales where the physical viscosity can finally act.

I suggest to read the fundamental textbooks about the vorticity dynamics and turbulence

Niko Niemelä

For me the easiest way of understanding the vorticity phenomena is to think it through the Reynolds number. When Re is large enough, the inertia of the fluid is so high that any small perturbation induces turbulence to the flow field. In other words, the internal friction of the fluid is not high enough to hold the flow field in structured laminar form, and therefore the turbulence emerges. Why the eddies are formed then? Think of a long stick (in vertical orietation) that is moving directly onwards with certain velocity. The bottom of the stick hits to an obstacle, the stick starts to spin around. Similar phenomena happens in the turbulent fluid flow when the shear stresses drag the fluid with different magnitude from different directions. In laminar flow the internal friction is high enough that it prevents the vortices to form, but in high enough Re number this is not the case.

And regarding the initial question: the fluctuating velocity components are caused by the vortical flow field (not the opposite as stated in the question).

Niko Niemelä

your framework makes sense if you have a turbulence that is related to the presence of body or walls with a specific characteristic lenght that allows you to define a ReL number.

But the general theory of homogenoeus isotropic turbulence has not a unique specific lenght to clearly identify the Re number. But also when you consider confined flows (e.g. channel flow) you have turbulence at low Re_tau number.

Actually in turbulence you can define a Re number for several characteristic lenght scales, for example at the Taylor or Kolmogorov lenghts the Re is very small.

Maybe is better to use the viscosity of the fluid to discuss the limit of the inviscid case and the inertial energy cascade.

Hamed Majidian

Dear Armin

Let me put it in this way, on the wall the viscosity is dominant relative to inertia, and thus viscosity resists against the translation of fluid particles. Going toward the free-flow region, conversely, inertia force is dominant and that is why the particles are prone to fluctuation even with a tiny excitation. However, particles from inner and outer regions of fluid change their layer frequently, while in the meantime due to momentum conservation they transfer another degree (higher or lower) of energy into the other layers, so leads to excitation, and due to mass conservation, they tend to return to their original layer. The outcomes of these phenomena end up with swirling in a fluid and generates eddies with different scales. By the way, the energy initially transfers from biggest eddies to the smaller ones, and it will be continued up to the dissipation of the energy in the smallest-sized eddies that possess the highest frequency or spinning rate. ( conservation of angular momentum= The smaller diameter is, the higher spinning rate is). At the end of the day, this chaos leads to turbulence in the flow regime.

Aaron Altman

Although not directly related to eddy production in turbulence, the following article provides an intriguing explanation for how and why vortices are formed from feeding shear layers. It is a classic (I believe Kaden was a student of Betz who was a student of Prandtl), but it builds on the argument that the eddies find their origins in viscosity

The turbulent flow is indeed characterized by spinning structures in the form of entangled vorticity tubes. When observing the evolution of the fluid, we can easily notice that these tubes meander, oscillate and change their connectivity in result of the process known as the vortex reconnection.

The mechanical model explaining the behaviour of the vorticity filaments was investigated by Da Rios in close collaboration with Levi-Civita. One of the first publication on the subject is available in Italian: L.S. Da Rios, Sul Moto d’un liquido indefinito con un filetto vorticoso di forma qualunque, Rend. Circ. Mat. Palermo 22 (1906) 117–135. where Da Rios derived the equations of the vortex filaments in the turbulent flow. The mechanical processes expressed in these equations answer the original question.

Levi Civitta continued work on the geometric model of turbulence, but unfortunately most of the results remained obscure for more than half a century and were only rediscovered in the 1960s. The theory is elaborated in the review article by Renzo L. Ricca (1996)

“The contributions of Da Rios and Levi-Civita to Asymptotic Potential Theory and Vortex Filament Dynamics, (Fluid Dynamics Research, vol. 18, pp 245-268).

This review also contains some references to modern publications but I think it is a good idea to familiarize yourself with the original contributions of Da Rios and Levi-Civita. It is perhaps important to mention that the renewed interest in the geometrical methods developed by these authors ultimately led to the topological fluid dynamics and to the description of turbulence based on the theory of knots (Moffat's articles are a excellent introduction to this area).

Mohsen Jahanmiri

Because of its diffusive and dissipative nature. It follows a cascade formation of eddies. No doubt that for this process to be continued it needs pumping of energy in a continuous manner.

Ed Gerck

AHK and all: The turbulent flow has energy to dissipate, and eddies offer an easy local way by rotational energy. The more incompressible the fluid, the higher the number of eddies. See also the reply by Janusz Pudykiewicz.

Helmut Ziegfeld Baumert

Turbulence is not just eddies, it is a very specific ensemble of vortices, still more specific: ensemble of vortex dipoles. Once generated somehow and equipped with a given kinetic energy, such dipoles can freely move like particles in the fluid and may form an Apollonian packing (Herrmann), the spectra formed this way were known already by Kolmogorov in the 1940th . What you mean with your "rotational energy", dear Ed Gerck? I am afrraid that also here Andrej Niokaljewitsh is the winner of the game: the second eqn in the two-eqn modls is NOT Epsilon, the dissipation rate, it is Omega, the r.m.s. gyroscopic frequency of the little gyroscopes formed by the vortex dipole filaments. Satisfied? This gives you all the fundamental constants of idealized turbulence. For details read UNIVERSAL EQUATIONS AND CONSTANTS ...

Mohsen Jahanmiri

“Because of its diffusive and dissipative nature. It follows a cascade formation of eddies. No doubt that for this process to be continued it needs pumping of energy in a continuous manner”

Your opinion is nicely illustrated by Lewis Fry Richardson in the following short poem:

“Big whirls have little whirls,

That feed on their velocity;

And little whirls have lesser whirls,

And so on to viscosity.”

― Lewis Fry Richardson

(The drawing is by Leonardo da Vinci)

Helmut Ziegfeld Baumert

“Turbulence is not just eddies, it is a very specific ensemble of vortices, still more specific: ensemble of vortex dipoles”

This observation is very similar to the idea expressed by Kelvin in the formulation of his theory of matter which later inspired those who applied the knot theory to studies of turbulent flow.

https://en.wikipedia.org/wiki/Vortex_theory_of_the_atom

Subhabrata Mishra

Eddy is nothing but a fluid current which has a different flow direction from the general flow direction. When the flow is in the turbulent region it produces eddies. If there is a flow and any obstruction comes into the flow path, there is a high chance of the formation of eddies.

As per your question, it forms due to the pressure difference. If you understand the formation of a tornado, you can understand this. Due to pressure difference at a point, fluid elements from all the region tried to travel and hence make a circular motion.

Better you search one-word "stagnation point". Then you think about the flow over a sphere. Then you may visualize the situation.

Dear all, but what does "eddy" really mean for you? Do you have a specific rigorous definition for it to be associated mathematically to turbulence? I have doubts about that... eddies, vortices, choerent structures, so many denominations are used in turbulence but a univoque mathematical definition is not addressed.

The only mathematical definition I can think of is the "vortical tube", that is the envelope of the vorticity field. But try to build a vortical tube in a turbulent field according to the Chorin's definition and look at the result....

Ed Gerck

All: This discussion is typical of current RG. It questions "eddies" but is not happy with a simple clarification ... whereas it is pretty obvious. The turbulent flow is THE energy source and has energy to dissipate, and eddies offer an easy local way by rotational energy. The more incompressible the fluid, the higher the number of eddies.

Ed Gerck

I am not sure to understand. In a statistically steady state you can immagine that production of energy equate the dissipation of energy in an integral balance. But that does not mean at all that all the "eddies" dissipate the kinetic energy by means of rotational. Furthermore, what does mean "the more incompressible the fluid..."??

Something in your statements is not clear to me

Ed Gerck

FMD: A compressible fluid can dissipate energy by compressing matter. An incompressible fluid only by movement, such as rotation, as in eddies....

Ed Gerck if you write the conservation equation for the total energy (compressible flow), the equivalent of the thermodynamic equations for a mass control,

dE/dt = W - Q

you see that the mechanical work has both a reversible and irreversible terms. The reversible part is due to the work of the pressure due to the volume expansion/contraction. I am assuming Stokes relation is valid. Actually, in compressible flows can be debated the adoption of a closure model for the isotropic part in terms of the classical eddy viscosity.

If you see the kinetic energy equation, the term p div v acts similarily. Dissipation of energy is expressed by (mu S:S). To be effective such terms you need an eddy on the small characteristic lenght approximatively defined by the Taylor micro-scale. Larger eddies (better to say vortical structures) does not dissipate energy but they "transfer" energy in an almost reversible way.

Have also a look to the spectral distribution of production and dissipation functions.

EG: try to find your loss words in a different forum. I immagine you have your own and solitary one to talk about with yourself.

Maybe you should be aware that all the matter is somehow compressible. "incompressible" and "compressible" are just models to be adequately defined depending on the character of a flow, that is the motion of a fluid a gas and so on.

Turbulent flows are non potential flows. Non potential means that the velocity field has non zero curl. In the turbulent state there is amplification of the vorticity by stretching, which explains why one observes vortices in turbulent flows. As noticed long ago by Kirchhoff many flows, like in the wake of obstacles are potential somewhere and non turbulent and turbulent in a bounded domain separated of the potential domain by a well defined boundary, not a material one. This splitting of flows in a turbulent and a non-turbulent part is explained in Landau Lifschitz course of fluid mechanics. It relies on the Kelvin theorem stating that circulation is convected by fluid motion.

Shahin Masoumi-Verki

Shear layers can cause Kelvin-Helmholtz instability which results in the formation of turbulent flow (please see two books of turbulence written by Lumley and Hinze if you look for a great course in turbulence).

As mentioned by Pope in his book of Turbulent Flows, turbulence is composed of eddies of different sizes (as we can see in the energy cascade). Now imagine you have a bulk of eddies, so you can have an energy spectrum for it. You cannot consider only one eddy in your works (sometimes it may be considered for defining some basics of the subject) but you have to consider a bulk of eddies. Therefore, a bulk of fluid or a bulk of eddies has the characteristics of the whole fluid flow, in terms of energy, since turbulence can be considered as a fractal phenomenon.

However, if you meant the motion of fluid flow and the formation of vortical regions in turbulent flow, you have to read about the forces on the flow (i.e., the stress tensor and its decomposition into two different parts: symmetric and anti-symmetric, and the motion that each of them can induce. Solid-body rotation and compression/tension). I recommend you to have a look at "An Introduction to Fluid Dynamics" by G. K. Batchelor if you want to understand the fluid motion.

Best wishes

Shahin

Felix Milan

"If you think you understand quantum mechanics, you don't understand quantum mechanics" -- Richard Feynman

If your question is "why...?" as in "how does a turbulent flow form eddies?" there's great literature available to dive deep into the subject, e.g. Tennekes & Lumley, U. Frisch, Pope and Davidson.

However, the way I understand your question is not really the "how?" but the "why?" on a physics "intuition" level. I see some some similarity in your question to Feynman's statement on quantum mechanics, referring (among others) to the interpretation of the collapse of wave functions. Also in the case of turbulence there's no intuitive reason (according to my knowledge) for a turbulent flow to form the structures referred to as "eddies". We have models to describe how they are formed, but not as to why. The Richardson cascade does not describe why they are formed but rather tells you how they "fit" into the picture of a multiscale dissipative flow. But there's no particular reason for such a flow to have eddies, i.e. there's no (theoretically inherent) reason for a turbulent flow to behave in the way we observe it to behave.

Or to say it in the words of Horace Lamb:

"... when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics and the other is the turbulent motion of fluids. And about the former I'm rather optimistic."

Thank you for raising this question!

TL;DR: We don't know why turbulent flows form eddies on a purely fundamental level.

Tatyana Belonenko

I agree with Felix Milan.

"We don't know why turbulent flows form eddies on a purely fundamental level".

There are many assumptions at the level of theories and with references to authorities. I have always been struck by this ability to speak without deep understanding.

However, we can see the formation of a vortex from turbulence in the basin. Please see the article.

Shchepetkin, A. F., 1995, Interaction of turbulent barotropic shallow-water flow with topography, 1995 Proceedings of Hawaiian Winter Aha Huliko'a Workshop, P. M\"uller and D. Henderson, editors, Honolulu, HI, pp. 225-237

http://www.soest.hawaii.edu/PubServices/1995pdfs/Shchepetkin.pdf

See Fig. 3 pls.

I think that the main reason for the formation of an anticyclonic vortex in the center of the basin and cyclonic circulation in the periphery is a topographical factor.

Tatyana Belonenko ,Formation of the vortical structures (eddies…) in the flow perturbed by the mountain is depicted in the attached animation showing the potential vorticity field. Vortices are indeed ubiquitous in the rotating shallow water flows. Please note the interesting example of the retrogression of eddies.

I am still waiting for someone giving us a "fundamental" way to define eddies... That would be necessary before stating that turbulence forms eddies...

Tapan K. Sengupta

I agree but, again, vorticity is not an eddy, it is "necessary" to define an eddy. So, what is the fundamental physical/mathematical definition of eddy? I never read about a univoque definition accepted by the community.

Could we talk about inertial vortical structures ? Should we add the coherence character? Should we exclude small vortical structures that rapidaly dissipate?

Tatyana Belonenko

Really, it is necessary to define an eddy.

And also to give a strict definition of the vortex boundary.

Is a vortex filament is an eddy or not?

Таня Шульга

I think that vortices arise due to the fact that the liquid is a continuous medium, the flow at the boundary captures particles with lower velocities, their acceleration is directed tangentially to the normal at each point, and then the velocities should lie on a circle.

Assuming the "eddy" associated to the rotational part (the topic of the Helmholtz-Hodge decompostion theorem is a bit more complex), assuming is a vortical structure, one could wonder why it appears also in laminar flows... Ok, you have single structures, not a cascade of eddies but the onset is also for laminar conditions.

And I agree, "this is related to perhaps answering what is turbulence".

Tapan K. Sengupta

and Readers,

The simplest answer to the original question is provided in Landau & Lifshitz Fluid Mechanics. In section 30, they identify the role of tangential instability in the generation of vortices, the analysis and definition of vortices are presented in section 31. It is really difficult to provide a more convincing and succinct explanation. I would perhaps only add the Ertel vorticity theorem to demonstrate how vortices are advected with flow and what is the role of a vortex stretching mechanism.

Vera Maura Fernandes de Lima

Tapan K. Sengupta

, thank you for your comment. The physical mechanism that leads to the creation of eddies is the dynamics of a vortex sheet expressed by an integro-differential equation proposed by Birkhoff in 1962, and Rott in 1956.

http://www.scholarpedia.org/article/Kelvin-Helmholtz_Instability_and_Roll-up

The alternative explanation if provided in the paper

“Instability patterns, wakes and topological limit sets”

https://arxiv.org/pdf/physics/0102005.pdf

where fundamental topological concepts are used to explain how the eddies are generated.

The vortex effect my friend. Read this:

Piralishvili S. A. Vortex effect (2020) A history of its development in the USSR and Russia AIP Conference Proceedings 2211, 020004 https://doi.org/10.1063/5.0001006

It arise from the interaction of a slow and a fast kinetics in the medium

Thank you for these suggestions Tapan K. Sengupta

. Chandrasekhar's book is a fundamental classic and I will check it for sure. This suggestion also brings an association with astrophysics and the turbulence in accretion discs. The questions about the mechanisms of formation of eddies is indeed fundamental. My experience with your book on numerical methods has been very positive and I look forward to reading your book “Instability of Fluid Flows and Transition to Turbulence”.

Vera Maura Fernandes de Lima

Vera Maura Fernandes de Lima Thank you for Your suggestion to consider the “Vortex Effect”. Unfortunately there is no open access to anything associated with this mysterious subject. I wonder if you can suggest any open access site to get some introductory information. Thank you in advance.

Dear Janus,

I thnk that the best contact to give you is Shota. He knows alot about vortices and also he is very nice and generous.

Shota A. Piralishvili here in RG

Thank you Tapan K. Sengupta

for your interesting comments and for the reference. The formal theory of atmospheric turbulence across the scales is still not complete. The most common conceptual model is based on the fact that the Earth atmosphere is relatively shallow, the basic description used in the theoretical analysis of mixing is the quasi-geostrophic turbulence, that is essentially two dimensional (2-D), as discussed by Charney in 1970s.

In 2-D turbulence we have conservation of both (kinetic) energy and enstrophy.

The energy spectrum in this turbulence has a well known form $E(k) \propto k^{-3}$. According to classical results of Fjørtoft there is the transfer of energy from small to large scales.

The opposite situation is observed in 3-D turbulence with a well known downscale direction of energy transfer from the large scale eddies, determined by the flow geometrical constraints and the distribution of heat sources, to the Kolmogorov inertial scale, where small eddies dissipate the mechanical energy and the spectrum is $E(k)\propto k^{-5/3}$. The theoretical analysis of the 3D atmospheric turbulence is still not complete because of the significant role of convection and the influence of orographic features.

I will try to present a classification of atmospheric eddies in the form of an Atlas to illustrate all these complex interactions.

Ijaz Durrani

Dear DR Armin : Simply put" In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object. "

In the lee of an obstacle, eddies form only when the flow around the obstacle reaches a critical velocity; they represent a flow of fluid into the space behind the obstacle, and this inflow begins only when the general flow is fast enough to produce a lowered pressure there.

Kind regards--

Vera Maura Fernandes de Lima

The generation of vortices in turbulent flows cannot be understood by simple arguments because there exists a solution of the Euler equation without viscosity for a flow around a sphere, without vorticity and without dissipation. This is known as d'Alembert Paradox and goes back to the mid eighteenth century. It is no simple matter to explain why vortices appear in the wake of a sphere in real flows at very large Reynolds numbers (up to 10^6 and even bigger) where viscosity should be negligible. The usual explanation by boundary layer does not explain why dissipation in this case follow Newton's law of friction like the square of the velocity. The correct explanation is that a localized and self sustained vorticity field is created in the wake by turbulent stretching of the vortex lines.

Dear Jannus I tried to follow you but from my chair here it looks that the theory you guys have is a big mess, a Frankenstein theorectical approach with a lot of room for improvement. Take heart the ecnomists andbiologist are in the same pickle place.

Yves Pomeau and Readers,

D’Alembert’s paradox can be resolved after analyzing the stability and physical realizability of a potential flow in questionable situations. It is very easy to verify that the zero drag potential solution is unstable with respect to small perturbations and quickly turns into a turbulent solution with very large drag. This instability is probably the main reason why the appearance of vortices is inevitable in the flow of the fluid.

Just as example of a similar concept, it is still somehow debated what produces instability and onset of turbulence in Plane channel. A viscous exact steady solution of the full NSE exists at every Reynolds numbers but the esperience says is not stable. And I still wait for a univoque explanation of the generation of vortical structures, the Re value for the onset of instability and the type of perturbation that produces transition...

What happens at the onset of instability in plane shear flows is well understood now. This is because the relevant instability is subcritical. It explains in particular that, at the onset, one observes, as predicted, spatio-temporal intermittency and that the transition belongs to the class of directed percolation. One has to go slightly beyond linear stability, a concept invented by Archimedes of Syracuse long, long ago!

Vera Maura Fernandes de Lima and Readers,

Thank you for your comments; Euler introduced his equations in 1757 with a laconic commentary ({\it ``nous aurons cinq equations qui referment toute la Theorie du mouvement des fluide''}) although he suspected that obtaining their solutions is not within the reach of his generation.

Today, we are still not able to fully comprehend the intricate behaviour of fluids, although some advanced mathematical methods will definitely lift this limitation. Perhaps the situation is the same in other disciplines including the biosciences.

The Euler’s paper is available on the web in the Euler Archive:

@article{Euler1757,

Author = {L. Euler},

Journal = {Euler Archive - All Works},

Title = {Principes généraux du mouvement des fluides},

Volume = {226},

Year = {1757}}

According to Chorin (he recalling Lamb..), one can define vortex line, vortex sheet, vortex tube. Each one implying a mathematical condition on the tengent property with the vorticity vector field. The term "Eddy" is somehow not related to a strict mathematical definition. Much more relevant is to define the character of "coherence" of a vortical structure. Ideed, the vortical structures within the inertial range have a coherence, differently from the small vortical structures in the dissipative range.

P. Contreras

you are wellcome, consider also that a hystorical references was published in JFM (1995):

Article Hussain, F.: On the identification of a vortex. JFM 285, 69-94

"have we digressed?"

I am not sure about ... the initial question is about eddy formed by turbulence, but could we talk about eddy formed also in laminar condition? For example is the laminar karman vortex shedding nothing but the realization of a coherent eddy? And again, are we sure to be on the same line about what is considered an eddy? Maybe arewe digressing somehow in the framework of a generalization of the topic?

Voskoboinick Andrey

Considerable confusion surrounds the longstanding question of what constitutes a vortex, especially in a turbulent flow. This question, frequently misunderstood as academic, has recently acquired particular significance since coherent structures (CS) in turbulent flows are now commonly regarded as vortices. An objective definition of a vortex should permit the use of vortex dynamics concepts to educe CS, to explain formation and evolutionary dynamics of CS, to explore the role of CS in turbulence phenomena, and to develop viable turbulence models and control strategies for turbulence phenomena.

Tatyana Belonenko

To Voskoboinick Andrey

Can you give a strictly academic definition of vortices as well as coherent structures? What is the difference between coherent structures and vortices? These questions follow from your answer.

Ijaz Durrani

Please take a look at the following link:

1-https://www.mpg.de/8252570/W002_Physics-Astronomy_042-049.pdf

2 on eddy motion:

https://www.sciencedirect.com/topics/engineering/eddy-motion

3- Eddie Formation and Turbulence:Article Eddy Formation and Turbulence in Flowing Liquids

Bijit KUMAR Kalita

In large scale circulations like ocean surface currents, eddies are formed due to the baroclinic instabilities. When density and pressure surfaces (contours) aren't aligned, then we have a baroclinic situation in the ocean and to get the "minimum energy state", eddies are formed as cyclonic(counter-clockwise flow in Northern Hemisphere) or anticyclonic (clockwise flow in Northern Hemisphere) circulations with barotropic (stable, stratified nearly laminar flow, which is very slow) background. Eddies are an efficient mode of water, heat, salt, and other chemical tracer transport in the ocean. Typically eddies are mesoscale features having 50-100 km of diameter, 4 weeks of a lifetime, and +(anticyclonic)/-(cyclonic) 5-10cm sea level anomaly.

Bijit KUMAR Kalita

That is interesting, however the physics you described is typical of the specific quasi-2D geophysical turbulence where merging and pairing of low scales turbulent structures can create larger scale single structure in a sort of inverse energy cascade.

The discussion about eddies in 3D turbulence is slightly different.

Dinku Seyoum Zeleke

You can estimate the eddie's size using Lagrangian time and length scale method by using turbulent fluctuation velocity.

Faraed Salman

In case of oceans, The equivalent of atmospheric storms, ocean eddies are spun off from currents. In this model, the Gulf Stream separates cooler water in the north from warmer water in the south. The Gulf Stream develops a loop that traps cold water in. This can completely separate to form a cold-core eddy.