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

As I have posted “Let’s Talk Turbulence – Part I“, explaining the qualitative universal nature of turbulence, one of my former appreciated professors at TAU sent me a remark: “But can you define turbulence?…”.

Well, easier said than done. The understanding of turbulent behavior in flowing fluids is one of the most intriguing, frustrating – and important problems in all of classical physics. Fully defining it is impossible.

But nonetheless as I love challenges, finding it better to fail a hundred than keeping dreams locked up in a safety-deposit box, I have decided to try and present different definitions I came across and perhaps link them by a scarlet thread to include most ingredients that should get as close as possible to somewhat of a definition – not formal though.

It is a fact that most fluid flows are turbulent and in many cases represent the dominant physics, on all macroscopic scales throughout the known universe: from the interior of biological cells, to respiratory and circulatory systems of living creatures, technological devices of modern age, to geophysical and astrophysical phenomena like atmosphere, oceans and cosmological physics reaching galactic scales and beyond.

*Notable outstanding results in turbulence research*

Saying all that, let’s start…

as a notion, I have added some remarks in italic for the original text whenever I felt necessary, sometimes when I felt it is in order for the scarlet thread to be formed.

- It is a well known fact that under suitable conditions, which normally amount to a requirement that the kinematic viscosity be sufficiently small, some of this motions are such that the velocity at a given time and position in the fluid is not found to be the same when it is measured under seemingly identical conditions. In these motions the velocity takes random (
) values which are not determined by the ostensible, or controllable, or ‘macroscopic’ data of the flow, although we believe that the average properties of the motion are determined uniquely by the data. fluctuating motions of this kind are said to be turbulent – G. K. Batchelor 1953**chaotic in nature seemingly random since NSE is a set of deterministic equations**

- The only short but satisfactory answer to the question “what is turbulence?” is that it is the general-solution of NSE (
) – P. Bradshaw 1972*the question of existence and uniqueness still stands*

- The distinguishing feature of turbulent flow is that its velocity field appears to be random and varies unpredictably. The flow does, however, satisfy a set of differential equations, the NSE, which are not random. This contrast is the source of much of what is interesting in turbulence theory (
) – A. J. Chorin 1975 (*an infinite dimensional in phase-space non-linear and chaotic dynamical system**fan alarm*🙂 )

- The next great era of awakening of human intellect may well produce a method of understanding the qualitative content of equations (
). Today we cannot. Today we cannot see that the water flow equations contain such things as the barber pole structure of turbulence that one sees between rotating cylinders. Today we cannot see whether Shrodinger’s equation contains frogs, musical composers, or morality – or whther it does not (*a plea for a 2D creature to acknowledge a third dimension*). We cannot say if something beyond it like God is needed, or not. And we can all hold strong opinions either way (*don’t try to answer that, you will be over-reaching…*) – R. P. Feynman 1963.**That’s why and correct**

- Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they flow past solid surfaces or even when neighboring streams of the same fluid flow past or over one another (
) – T. von Karman quotes G. I. Taylor.*That’s a description rather than a definition – Turbulence happens*

- Das “Turbulenzproblem” der Hydrodynamic ist ein problem energetischen, nicht der dynamischen stabilitat (
) – W. Heisenberg 1923.*yes, it’s german, and it is somewhat correct but not entirely otherwise we wouldn’t take the route of stability theory for transition research*

- Creation of small scale activity and dissipation, is the principle of turbulence. Classical fluid dynamics instabilities play a role of the fuel, vortex stretching is the engine (
), and viscous dissipation is it’s breaks (**so is vortex tilting**) – P. Constantin 1994.**but how does it sustain?… still a lovely description**

- Turbulence can be defined by a statement of impotence reminiscent of the second law of thermodynamics: flow at sufficiently high Reynolds number can not be decelerated to rest in a steady fashion (
). The deceleration always produces vorticity, and the resulting vortex interactions apparently so sensitive to initial conditions (**circular causation alarm**) that the resulting flow pattern that changes in time and usually in a stochastic manner (*Chaos we said…*) – H. E. Liepmann 1979.**stochastic is a very appealing replacement to ramdomness which is somewhat incorrect**

- A body of fluid is a mechanical system with an infinite number of degrees of freedom.v It may therefore be expected to a execute a (
) random motion comparable to that of molecules in gas. If one regards such a chaotic motion as analyzed into harmonic components of variation of scales , one recognizes tend to dissipate the small case oscillations and keep the motion more less regular. Thus, when viscous forces are sufficiently strong, i.e.at sufficiently low Reynolds numbers, the motion wail become laminar. On the other hand, at sufficiently high Reynolds numbers the motion will tend to become (seemingly) random fluctuating, even when when external conditions are steady (**seemingly**) – C. C. Lin W.h. Reid.**again the question of sustained turbulence cycle and the Reynolds number dependency – remembering that essentially the Reynolds number is a dimensionless number reflecting on how well momentum is diffused relative to the flow velocity in the cross-stream direction)and on the thickness of a boundary layecr relative to the body – it’s hard to drw a deffinition for turbulence by such**

- We have therefore defined turbulence as as random fluctuations of the thermodynamic chrecteristics of vortex flows, thereby distinguishing it it at outset from any kind of whatever irritotational, i.e. potential flow,… (
) – A. S. Monin (Fan Alarm 🙂 )**somewhat of a circumvention, but it seems that avoyage for the definition of turbulence could be achieved upon directing the phrase, if it were possible**

- Perhaps a satisfactory definition would be an ensemble of non-periodic solutions to the NSE. Ensemble of solutions simplified or otherwise modified forms of th NSE will not qualify as turbulence, we shall regard them as models of turbulence (
) – E. N. Lorenz 1972.**This is a huge step forward and it is not a Lorenz quote by chance, it’s quite amazing he published it in 1972**

- Turbulence is the name given ti imperfectly understood class of chaotic solutions to the NSE in which many (and denied a spectral gap) degrees of freedom are excited (I f we call a dog’s tail a leg than how many legs does a dog have?… It’s still four!) – H. Aref 1999.

- Turbulence is a three-dimensional, time-dependent motion in which vortex stretching (also tilting and fractalization) causes velocity fluctuations to spread to all wavelength between a minimum determined by viscous forced and a maximum determined by the boundary conditions of the flow. It is the usual motion except at low Reynolds (
) – P. Bradshaw (Fan alarm 🙂 ).**Still to much reliance on the Reynolds number, but a general definition which enables to capture an obscure definition**

- Turbulence is the name given to imperfectly understood class of chaotic solutions to NSE (
) in which many degrees of freedom are excited (reminding the spectral gap issue). H. Aref 1999.**they are know to be chaotic by nature, proof of their uniqueness is still mandatory and it cannot be drawn by DNS due to the chaotic nature and the sensitivity to seemingly identical initial condition and the numerical/experimental error**

- Turbulence with it’s limit of self excitation, with the characteristics hysteresis in its appearance and disappearance and appearance as the velocity of flow produces as the velocity of flow producing it is increased or reduced and the primary role of non–linearity in its development (stationary) state, is,in fact, a self oscillation (a hint for the self-sustained process).tfa Its specific features are determined by the fact that it is self oscillation of a continuous medium, i.e., a system with infinite degrees of freedom (I appreciate this description especially) – G. S. Gorelic.

- Before 1970 I wouldn’t have dream of putting the words turbulence and predictability side by side… . To me turbulence was unpredictable by definition. Turbulence was the chaos that arises in fluids because of innmerable instabilities associated with vortex strtching (
)**tilting and fractalization**

These days, I tend to think turbulent flow qw flow in which deterministic calculations become useless over a finite interval. – H. Tennekes.

- Everyone who , at one time or another, has observed the efflux from a smokestack has some idea about the nature of turbulence flow. How ever it is very difficult to give a precise definition of turbulence. All one can do is list some of the characteristics of turbilence flow:

Irregularity… Enhanced Diffusivity… Enhanced dissipation… Large Rynolds numbers, Three-Dimensional, Vorticity fluctuation and their obvious strtching, tilting and fractalization Mechanisms,Continuum, Chaos…………… Turbulence flows are turbulence flows – H. Tennekes and J. L. Lumley (A first course to turbulence, MIT – Extremely recommended)

I can continue almost forever. But these quotes satisfy me

So I will close this chapter leaving the reader waiting for my publication of “Define Turbulence… PART II”, where I shall collect the the knowledge of these seminal quotes of above along with my notes (to my best understanding) and mind my self not to fall from the height of the shoulders of such giants while forming carefully my scarlet thread linking those wonderful quotes as close as can be to a definition of turbulence.

Here is a chllenge for my readers, why don’t you take part and explore your scarlet thread to your own understanding of the definition of turbulence…

### A book extraordinary recommendation from a CFD blogger:

Some more recommended CFD readings:

## This blogger favorite book list

A must read for turbulence modeling understanding:

(Including with F. Menter references)

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