In order to be accurate one needs to distinguish that Chaos has a very accurate MATHEMATICAL definition, while Turbulence is a PHYSICAL Regime of Fluid Flow, without an accurate MATHEMATICAL definition.

The typical characteristics of Turbulence have been mentioned in the answers above, but not all of them together. Irregular fluctuations in time as well as in space, small as well as large three dimensional eddies (vortices) coexist. Whether these are stochastic or chaotically deterministic has never been proven (as far as I know). So, Turbulence might be one example of the physical manifestation of deterministic Chaos, OR a stochastic, non-chaotic, manifestation of the solution to the nonlinear fluid flow problem at high Reynolds numbers. There is no conclusive PROOF of either of the alternatives listed above.

Please read this artticle: "ORDER AND CHAOS IN FLUID FLOWS" by R Narasimha - Current Science, 1987 - repository.ias.ac.in

Turbulence are the result of from organized convection as warmer air rise in one column of air while colder air sinks in the adjacent column of air. This pattern is repeated over an volume of space in an area of turbulence.

Chaos has no pattern, organization, or any defined reasoning of its existence.

This article may be helpful

A chaotic system is a system where small perturbations of the system conditions (e.g. initial conditions) result in very different system behavior.

The word turbulence is used to qualitatively describe the behavior of fluids or fluid-like systems where energy is dissipated in vortices. Turbulent properties are definitely stochastic but this is not the same as being chaotic. For a turbulent system to be chaotic, I would e.g. require the turbulence statistics to change considerably by performing small perturbations of the system properties.

Furthermore, non-fluidic systems may behave chaotically. E.g. in a three-body system subject to gravitation only, a slight change in the distance between the bodies will result in a completely different pattern of movement. Thus the three-body system is a chaotic one. A two-body system, however, is not.

To summarize; chaos is a word used to classify the sensitivity to perturbations in dynamic systems while turbulence is a manifestatition of fluid flow-property stochastics.

Dear Mohsen Jahanmiri.

Unfortunately the article is reserved for the working group, there's no way to access it.

Anyway, thanks for the suggestion.

In order to be accurate one needs to distinguish that Chaos has a very accurate MATHEMATICAL definition, while Turbulence is a PHYSICAL Regime of Fluid Flow, without an accurate MATHEMATICAL definition.

The typical characteristics of Turbulence have been mentioned in the answers above, but not all of them together. Irregular fluctuations in time as well as in space, small as well as large three dimensional eddies (vortices) coexist. Whether these are stochastic or chaotically deterministic has never been proven (as far as I know). So, Turbulence might be one example of the physical manifestation of deterministic Chaos, OR a stochastic, non-chaotic, manifestation of the solution to the nonlinear fluid flow problem at high Reynolds numbers. There is no conclusive PROOF of either of the alternatives listed above.

The article suggested by Jahanmiri can be downloaded from the journal's website.

The following link downloads it directly:

http://www.currentscience.ac.in/Downloads/article_id_056_13_0629_0645_0.pdf

The full reference is:

Narasimha, R., Order and chaos in fluid flows, Current Science (A fortnightly journal), Volume 56 - Issue 13, 5 July 1987.

Sverre e Valdsz.

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Thank for responsible answers, showing the seriousness with which they were written, because for many seems simple definition of the two concepts, and show you what is not.

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I put this question just to make clear to most people that should avoid the frivolous use of the concepts of Chaos and Turbulence, something that is made in many peer reviews articles in important journals.

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The last sentence of Vadasz resume the uncertainty which have about such concepts:

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“So, Turbulence might be one example of the physical manifestation of deterministic Chaos, OR a stochastic, non-chaotic, manifestation of the solution to the nonlinear fluid flow problem at high Reynolds numbers. There is no conclusive PROOF of either of the alternatives listed above.”

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I think it is clear to all that chaos is a mathematical description of the behavior of physical systems with finite number of degrees of freedom or not ("or not" I still have my doubts). On the other hand turbulence reflects a state of disorder of a physical fluid flow.

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I see that we can still stay with the old adage that I learned back in the 80s with Professor Rouse, “You should not be defined turbulence, you must characterize it.”

No....there is no clear separation between chaos and turbulence but chaos is the stage between order and disorder while turbulence is the state of disorder.

Actually, chaos is already the state of disorder, just as turbulence. My statement above did not refer to "separation" between chaos and turbulence, but rather about the existence or lack of a DEFINITION.

The zeroth law of turbulence is that the kinetic energy dissipation rate tends to a non-zero limit as the viscosity of the fluid, \nu, is reduced to zero i.e., the limit \nu \to zero is singular. Analogous statements, amounting to the existence of a singularity or dissipative anomaly, can be used to define or characterize "turbulence" in non-hydrodynamic contexts i.e, time reversal symmetry is not restored in the limit in which the symmetry breaking parameter goes to zero.

This may not be a fully fledged definition but it does serve as a necessary condition for macho turbulence, as distinct from mere disorder, sensitive dependence on initial conditions, nonlinearity etc.

Commonly chaos is defined a low dimensional nonlinear dynamics described by a nonlinear ordinary differential equation (ODE) (see Lorenz or Roessler equation). Turbulence is described by a nonlinear and nonlocal partial differential equation (Navier Stokes)- corresponding to a very high dimensional system. Expressing turbulence by ODEs one needs many degrees of freedom - about Re^3 (Re- Reynold number). Both chaos and turbulence have some common features as sensitive dependence on initial conditions. Big simplifications of the turbulence equations lead to chaotic equations. A common scenario from convection is that the dynamics will change from - laminar - oscillatory - chaotic - spatio-temporal chaos - weak turbulence to fully developed turbulence .

William: the zeroth law is "not even wrong" but has generated much disorder, wondering, speculation, guessing etc. in the world of turbulence theoreticians and modellers. However, it is much less a monster than it looks like. It even gave way for a universal solution of the (asymptotic) turbulence problem, (A), see below and attachment.

Joachim: when I mention a solution of the (asymptotic) turbulence problem (A), then this tells nothing about the second part of the turbulence conundrum, (B): the transition. Here I believe with all respect that Prandtl (and many others) were on a wrong path when they believed in potential universality. The details matter too much here. You mention the case of convection and I agree. Other routes you find for shear flows and stratified shear flows, not to mention the multitude of non-Newtonian fluids, a topic of increasing practical relevance but only limited understanding. One might conclude that "the turbulence conundrum", consisting of (A) universal laws for Re => infinity and (B) laws of transition, can only be solved for part (A), never in a universal way for (B).

Rogerio: the answer to your question is YES. Let's talk about turbulence in the narrow sense of fluid dynamics. The spectrum of kinetic-energy fluctuations in a turbulent flow exhibits two major scales, the so-called energy-containing scale, L, and the microscale, lambda, named often after Kolmogorov. Turbulence in a scientific sense is bounded by these two scales and follows Kolmogorov's 5/3 law. The integral over this interval gives the turbulent kinetic energy, or TKE. That's it. In the attachment you may find the case where lambda is zero and Kolmorov's wavenumber thus infinity.

Last week I just gave in Tallinn at IEEE 2014 a generalization of the theory wich works for finite Reynolds numbers and thus real-world turbulence, even for (dilatant) non-Newtonian fluids. The classes of motions and associated kinetic energies outside the strict turbulent interval (ie. below lambda and above L) have surely stochastic character, they possibly even may be called "chaotic" -- I don't know and never made much intellectual profit from the notion "chaos". However, the scalar mixing capabilities outside the TKE interval is weak. Maximum mixing per kilowatt you get from a high ratio between TKE (within L and lambda) and total kinetic energy. If L = lambda, then mixing even collapses completely.

Dear Helmut.

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Thanks for the article, I will look with the care it deserves. If conditions'll find me some comments.

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Dear Rogerio, any comment with page and equation number is welcome!

The physical manifestation in a three-dimensional structure reacting to energy inputs over time is subject to level and rate of change of energy input, degrees of freedom in the system, and initial conditions.

At low energy inputs, the manifestation may at first be linear, moving toward linear flows centred on an attractor (i.e: convection).

At higher energy levels, small changes in initial conditions can create bifurcations in the linear pattern (via three dimensional fractalisation), such that non-linear systems arise at the quantum level, leading to flip-flop of pattern changes towards different attractors at different energy levels, leading to alterations in the pattern of the overall structure. This is what I visualise as the CHAOTIC manifestation.

At energies above this set of extremely sensitive initial conditions, the chaotic manifestation breaks down, the ordered bifurcations no longer applying, and disorder in a system without direction ensues, in that there is no longer any group of similar attractors strong enough to maintain a semblance of order in the system.

The system therefore breaks down into multiple sub-units, each trying to maintain their own hold on separate attractors within the matrix, each sub-unit comprised of varying degrees of freedom, and localised initial conditions. This is what I visualise as TURBULENCE.

Dear Robert, you describe turbulence in a wider sense. Today we know more about turbulence and may define it in a more narrow, specific sense in terms of r.m.s. observables, i.e. in terms of energy, r.m.s. vorticity, eddy viscosity of momentum and scalars (and more). That's what the engineer wants to get for given boundary conditions and that's what we may deliver based on our universal theory if the Reynolds number is "high enough", Re > 10^4 or better. Remains to be specified: the transition to turbulence from laminar motions. Here we have no universality but various analogies with dynamical systems theory, as you wrote.

Waleed,

turbulence contains both kinds of motion in a continuous wavenumber spectrum: the low-wavenumber part is typically described in terms of chaotic dynamics and strange attractors, i.e. as solutions of low-dimensional non-linear systems - as mentioned above by Joachim. It strongly depends on the Reynolds number and reaches until the beginning of the inertial or Kolmogorov part of the spectrum. This point  is called the energy-containing scale.

The latter inertial spectral part reaches until the Kolmogorov wavenumber (in the best case: \infty) characterizing the microscale where turbulent kinetic energy is dissipated into heat. The inertial (or Kolmogorov) range is smooth and characterized by the 5/3 law, first formulated by Kolmogorov. It reflects kinda "devil's clockwork" or "devil's gear" proposed as a mechanical analogy by Hans Herrmann.This spectral part exhibits a well-ordered spectrum although the processes are essentially of stochastic nature.

Today in geophysical fluid dynamics the low-wavenumber part of turbulence is called EKE = Eddy Kinetic Energy (in contrast to TKE =  Turbulent Kinetic Energy) and studied eg. by Carsten Eden and Dirk Olbers - if I am not in error.

The difference between EKE and TKE: EKE is only weakly mixing scalars like heat whereas TKE is the real mixer: the "devils clockwork" mixes scalars down to the Kolmogorov microscale. I.e. heterogeneous patches may stil have only the size of the microscale where further molecular mixing takes place down to molecule scales.

As the mixing in the EKE part is negligible, mixing is generally governed by the so-called energy-containing scale only.

Waleed,

it is correct that a chaotic system is deterministic. E.g. the three-body system subject to gravity, only, is definitely determinstic, since the three bodies' position and velocity is only given by their initial conditions, and no random effects take place. I.e. if you know positions and velocity vectors of all three objects at any point in time, you may calculate their respective positions and velocity vectors at any other point in time, before or after. Chaos may, however, render the determinicity of the system useless for predicting its future state, for all practical purposes, due to the complexity of the motion pattern.

Even if a chaotic system is deterministic, statistics may be useful to describe it. E.g. for a dilute gas, statistical physics and thermodynamics is more useful than calculating the motion paths of each inidividual gas molecule, although this is in principle, possible, if the initial condition is known.

A stochastic system, is subject to chance (or probability if you prefer). It is not possible to calculate the future (or past) state of a stochastic system based on knowledge of its present state. You can only calculate the probability that it will be in a given state. This is due to random effects taking place.

Dear prof. Rogerio,

My dissent contribution to the discussion.

"Is it turbulence order instead of chaos?"

From the second law of thermodynamics, it has been proven that there is only one way to sustain order within a material system: it must be open to energy flow otherwise entropy (or chaos, as you prefer) will inevitably grow and eventually obliterate the structure of the system.

My point is that near the edge of transition to turbulence regime we can find the optimal state for the rise of a self-organized material system.

To illustrate that, see Rayleigh-Bénard convection cells. My main reference: BERGÉ, P.; Rayleigh-Benard Convection in High Prandtl Number Fluid, Chaos and Order in Nature - Proceedings of the International Symposium on Synergetics at Schloll Elmau, Bavaria, April 27 - May 2, 1981

best regards

http://www.bookmetrix.com/detail/chapter/619585a0-af4d-4dec-b600-2271ab1b8374#downloads

Chaos is deterministic, meaning a set of equations can be used to describe the behavior of the system for given initial and boundary conditions. However, the exact nature of the outcome depends on how accurate the initial condition are as these set of equations are rather very sensitive to these initial conditions.

While turbulence is random and we cannot express its behavior without a suitable "model". This is known as the "closure problem". The similarity is that the NS equations are very sensitive to initial conditions are (was) believed to have a limit cycle sort of solution. You can refer the introductory chapters of " A First course in Turbulence" by T and L or Stephan Popes Book "Turbulent Flows" for further queries.

My turbulence professor ( former student of Prof. Lumley) often brings in an anology from links and chains (in context to degrees of freedom). For a simple chain with 2 links hanging vertically, the degrees of freedom are easy to estimate, but a rope (similar to turbulence) has infinite degrees of freedom!