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 sakti September 21, 1998 02:33

eddie &vortex

hello everybody

I'm a student. I'm interested in turbulence. I'd like to ask about : what is the difference between vortex and eddie in turbulent flow?

what is the difference between random and chaos (in case of propeties of turbulent flow)?

if we do not know the critical reynolds number, can we define a flow field is turbulent flow, only from the picture (velocity vector)that had rotasional structures ?

thank you all very much

sakti

 Jonas Larsson September 21, 1998 05:46

Re: eddie &vortex

Your questions are bordering to theoretical turbulence, which is not really my area of expertise, but I'll answer you anyway. I hope someone else can fill in and correct me if I miss something.

1. Difference between eddie and vortex.

When you talk about "eddies" your most often mean to turbulent eddies - small localised vortical stuctures that you commonly find in turbulent flows. A vortex can just as well be a large scale structure in the flow and need not be related to turbulence. The two words often have the same meaning though.

2. Difference between random and chaos.

There are long books just about this. I don't like when people use the word "random" in a loose way as a definition of turbulence. Turbulence is not random, it is a fully deterministic fluid motion which can be predicted provided that you have a fast enough computer and know all initial and boundary conditions, at least that is what I belive. There are, as far as I know, no rigorous mathematical proofs of either existence or uniqeness for Navier-Stokes in general. So the question is still in some sense open for debate I guess.

When people talk about "random" in relation to turbulence they usually refer to the fact that at a large scale the small turbulent motions appear to be random and unpredictable. This "scrambling" of information is not randomness, although it can appear so, it is an important chracteristic of turbulent flows though.

Chaos is something else. Turbulence is indeed chaotic and a small change in initial condition can change the final result drastically and this is also very difficult to predict (but it is theoretically possible).

Random = the output from a theoretically unpredictable process (like the decay of an atom, quantum leaps, ....)

Chaotic = non-linear, complex, irregular process which is difficult to predict (non-computable even).

Chaos can often falsly appear random if you look at it at a large scale. Does randomness exist in reality or is it just an illusion because we are looking at things from the wrong perspective? Most physisist will answer yes, we have randomness in the world, but many will disagree with this.

3. define a flow field is turbulent flow

Every book about turbulence seems to have their own definition of what it is. There is as far as I know no strict theoretical definition which holds in general. A good way to define turbulence is to describe some of its properties like: irregular and chaotic flow (often falsly called random), occurs at high Reynolds numbers, includes 3D vortical structures and 3D vorticity variations, is highly diffusive, transports energy in the sence that it takes energy from large scale structures and dissipates it at small scales, it is a continuum phenomena.

Hope that helps

 R.D.Prabhu September 21, 1998 23:24

Re: eddie &vortex

Just an addition to what Jonas has explained:

eddy : Turbulence is NECESSARILY rotational. We often use simple fourier series to decompose any random field.keeping the two in mind , we often speak about each fourier mode(k)as 'eddies' on that scale (~ 1/k)

vortex: A large body of fluid in rotation for a sufficient length of time . This picture is more physical than that of an eddy. The lifetime of such coherent motion is more than the standard 'eddy turnover time'

Random: As Jonas puts it in the spirit of Boltzmann, given the Laws of Physics and the intial conditions of 'EVERYTHING' in the UNIVERSE, we can in principle PREDICT the FUTURE!!! ( no randomness here) But since the problem is bogged down by it's sheer weight of DIMENSIONS ( forget about the`universe'.. a fluid itself is regarded as an INFINITE dimensional system!!) we need to develop the science of RANDOMNESS.

Say..any event that resembles COIN-TOSSING in terms of it's distribution of all possible outcomes is safely RANDOM.

Chaos : there is as of now no general consensus about the definition of CHAOS. But there are certain WORKING Definitions...in vogue..

1. CHAOS IS a specialized case of Randomness. What is special is the systems SIC'ness! i.e..Sensitivity to Initial Conditions.

2. this sensitivity IS characterized by an exponential sense of 'divergence'( of two nearby trajectories in phase space). hence technically.. any system with POSITIVE LYAPONOV exponents IS CHAOTIC.

3. Such systems invariably exhibit the presence of a 'zone' in phase space where most of the trajectories get 'attracted' to. and this ATTRACTOR cannot be described in the EUCLIDIAN sense..( say ..cubes, circles, or spheres..)..In short it is a STRANGE- ATTRACTOR..( the world of FRACTALS starts from here!!)

Hence Chaotic systems ARE characterized by the presence of STRANGE ATTRACTORs . MIND you the converse need not be TRUE..viz..the presence of a strange attractor need not imply CHAOS.

4. finally NON-LINEARITY is the KEY..(not multi-dimensionality as in the case of Randomness) even a low-dimesional NON-LINEAR system may exhibit CHAOS.

well is that the end of it..? no.. ask WHAT IS TURBULENCE ?? again .... It's not simply CHAOS ..it is a case for SPATIO-TEMPORAL CHAOS ..( all that we were talking of till now was just TEMPORAL.. include SPACE now ...and see what a MESS it becomes) Hope this DOES (not) confuse!

Prabhu

 Jonas Larsson September 22, 1998 02:41

Re: eddie &vortex, does true randomness exists?

Getting a bit off topic here, but anyway, I disagree with you, the universe is not deterministic, and there is a clear distinction between random and chaotic. The general concensus among most of todays physicists is also that many processes are random by nature. It is not just that we can't see the order behind the chaos. Fluid dynamics is deterministic and "non-random" though.

The question of whether or not the universe is computable (predictable) is different. A random universe can very well be computable, although the end result will include probability distributions. And a deterministic universe, if such a beast does exists, can also, as you say, be non-computable.

 R.D.Prabhu September 22, 1998 16:11

Re: eddie &vortex, does true randomness exists?

Well, the Universe is deterministic in the Boltzmann'ian sense as I said earlier. ( classically) If you believe in the `wave-function' of quantum mechanics then it is a little more subtler a view-point that ... ( which is still being debated) randomness will have to be hypothesized as the 'ROOT' of everything.

And I do believe that we have this well-developed language of Randomness even for the quantum world..viz..expectation values, transition probabilities, etc etc ..

No wonder Feynman ( as quoted by Uriel Frisch) describes Schrodingers time dependent equation for the wave function ..as ..the 'equation for LIFE'!

coming back to fluid mechanics..do you think that TURBULENCE can `purely' be produced from Navier-Stokes alone? ( since you said fluid dynamics is deterministic)

Prabhu

 Jonas Larsson September 22, 1998 16:46

Re: eddie &vortex, does true randomness exists?

Yes, turbulence is a continuum phenomena and it can be fully described by the Navier-Stokes equations. There is no randomness there.

Many turbulence books use the word "random" to describe iregular and chaotic flows. It can be misleading because people confuse this with true random physical processes (decay of atoms, ...).

 R.D.Prabhu September 22, 1998 22:46

Re: eddie &vortex, does true randomness exists?

But Jonas,

There is a more fundamental way of looking at fluids.Treat the continuum as a collection of particles. Write the more general boltzmann equation for these particles. you still are left with the collison terms ..two body, three body, n-body you stop with a model somewhere, right..? the moments of this equation is what is Navier-stokes. More fundamentally treat each particle as a wave-packet, attatch a wave function for your whole system, write the Hamiltonian and plug it in schorodinger's equation, and solve!!

sorry for the mess but more crudely I feel that the Navier-stokes equations are NOT the EXACT description of fluids. (don't we model our viscous-stresses with Newtons's law..?)

Prabhu

 Zhong Lei September 23, 1998 00:50

deterministic differential equaitons = a deterministic solution?

The problem is whether we can always get a deterministic solution from deterministic differential equaitons, where initial and boundary conditions may be deterministic or random.

I do not know whether DNS (Direct Numerical Simulation) of turbulence and some theories of quantum mechanics are a answer.

 Jonas Larsson September 23, 1998 03:16

Re: eddie &vortex, does true randomness exist?

I agree with everything you say, the Navier-Stokes equations are not exact. However, your question was if turbulence can "purely be produced from Navier-Stokes alone". The answer to this question is yes. I guess you can make an argument that small random pertubations originating from quantum effects could affect the turbulence. This effect is however negligeble - compare the Kolmogorov micro-length-scale to the typical size of a molecule if you aren't convinced. The micro-scale of turbulence is many orders of magnitude larger. We can already neglect molecular effect in most type of flows, and quantum effects is one (or rather two) steps further down in refinement. There would be turbulence, and it would behave the same as it does now, also if we lived in a perfect "classical physics" world.

The use of the word "random" in turbulence leads to just this kind of missunderstandings. Lets call it "irregular" and "chaotic" instead.

 C.,H.,Han. September 23, 1998 08:57

Re: eddie &vortex, does true randomness exist?

Speaking publically with a small knowledge will annoy you. But I have an opinion and want to write it, even though I am to be blamed.

---------------------------------------- " sorry for the mess but more crudely I feel that the Navier-stokes equations are NOT the EXACT description of fluids. (don't we model our viscous-stresses with Newtons's law." -----------------------------------------

Dear R.D.Prabhu,

There exist phenomena. And we try to express those phenomena with some equations derived by modeling them. N-S eq. is another version of Newton's 2nd Law by some assumptions, like continum hypothesis, nonlinear or linear relations between stress and change rates of strain,...etc.

We treat turbulences as fluctuation terms with a statistical view, so we derive RNS eqs with some averaging procedures.

Only different in what way we describe phenomena.

And now,

Is the world deterministic or probabilistic? If deterministic, we can find a solution from N-S eqs. If not, then we can also have an appromximate solutions with some treatments.

Chaotic means deterministic, even though difficult to predict? or means probalistic , expressed wrongly by some treatments of deterministic view?

The N-S eqs. are based on Newton's 2nd Law with auxiliary eqs. from assumptions Since Newtonian view is deterministic, N-S eqs. are somewhat misleading if the world is probalistic.

You said the universe is deterministic. In the quantum level, I don't know exactly. But I think in the macroscopic level, deterministic.

The Bottomline:

N-S eqs. can give a solution by specifying some physical conditions.

---------------------- In Cruel Reality, --------------------

You said about generallized-type Boltsmann eqs. I think if we have a really fast COMPUTER, we can solve it in the quantum level with some generalized eqs for the infinite number of particles. But we can't solve it because we don't and will not be able to have a fast computer. So, if possible, we have to find approximate solution with error bound and it is good for some engineering purposed if only we can have good approximated solutions - it is really difficult to have them over complex geometries, flow phenomena .

 John C. Chien September 23, 1998 12:10

Re: deterministic differential equaitons = a deterministic solution?

1). The summer is no longer with us. As you walk up to a little pond in an early morning, there's no wind, and the surface of the water is mirror flat. You ask yourself,"which word should I use ?". The answer seems to be " Random". As long as there's no wind, no fish, no bird, the surface is mirror flat, the only way the water temperature can remain at 68 degree is because of the "random nature of H2O molecules in the pond". The number of molecules in the pond is "Huge!". You don't want to count them one-by-one. And there's no way of knowing their initial conditions. Well, still you may want to sit under the apple tree, to read the kinetic ( molecular) theory of gases, or watching the mirror flat surface until it becomes " Chaotic". It's interesting to observe that " mirror flat" does coexist with " Random " nature. Perhaps, the " Scale" is an important factor. 2). It's hurricane season now. If you happen to live in the south-east coast of North America, you are likely to see a counter-clockwise rotating hurricane on a TV screen at news-time every night. It moves only a few inches on the TV screen every 24 hours only,yet, people are very serious about its movement. "W", "NW", "NNW" seem to make a big difference. The picture taken from satellite always never tell the truth about the "turbulent, chaotic" nature of the hurricane. It's predictable in the news, yet it's highly chaotic. If your house is in the hurricane path, your roof could end up in your neighbor's backyard or it could sit on the street 100 yard away. In this case, no one will argue with you about the turbulent nature of a 100 miles per hour wind. So, the two versions of the hurricane seem to be related to the definition of " time-scale" and " distance-scale". U(x,y,z,t), V(x,y,z,t), and W(x,y,z,t) alone do not seem to produce useful answer at all unless you include the distance-scale ( length-scale ) and the time-scale into the picture. The random nature of a mirror flat pond water surface in the early fall season morning is simply quiet and beautiful !

 R.D.Prabhu September 23, 1998 15:20

Re: eddie &vortex, does true randomness exist?

Yes, Let me elaborate ..why I doubt that N-S can by itself produce turbulence ( although for all practical purposes our discussion makes no difference to the engineering world of applications).

Correct me if I am wrong,

1.) Experiments on transition to turbulence have revealed that the critical Reynolds number is sensitive to the roughness of the surface. ( doesn't that mean turbulence is excited by these `external' perturbations !?)

2.)Although DNS has reproduced kolmogorov law for us, even here don't we start with an intially random field..?

3.) Starting from a laminar state then , (with just one mode k), can a turbulent field be produced which has similar statistical resemblance as our experimental ones..? To give it a historic touch this is what GI Taylor asks in his conversations with Batchelor. Well, I do not know if this question has been answered.

now, coming to quantum fluctuations and origin of turbulence, I think it is inappropriate to discount quantum effects using scaling arguments as you suggest, in the light of our question. WE are asking what CAUSES turbulence, not what are the different regimes once there IS turbulence, right..? To me this looks like saying "since the Universe is SO BIG, we need not worry about quantum effects to understand it's ORIGIN."

Prabhu

 R.D.Prabhu September 23, 1998 15:37

Re: eddie &vortex, does true randomness exist?

Dear Han,

I don't deny that our intelligent approximations ( say N-S equations) WORK. For all practical purposes THEY ARE indeed very effective and helpful . Our language of understanding the world around us is no doubt near PERFECT.

However when we like to be MORE EXACT, the problem starts. We fall short of identifying the TRUE CAUSE in such cases. No doubt our power of approximations is unquestionable.

Regarding the UNIVERSE ( without digressing much) so far quantum mechanics rules in our understanding. But we still do not completely understand the implication of QM ( to be a little technical ..say 'collapse of a wave function' when the act of 'observation' is made!!). So , the universe is best described probabilistically!

ALthough Penrose would elaborate further along these lines and explain the inadequacy of QM too!

Prabhu

 R.D.Prabhu September 23, 1998 15:43

Re: deterministic differential equaitons = a deterministic solution?

In short,

Beauty lies in the eyes of the beholder (his/her own time and length scales), and the whole world is RELATIVE.

well said sir, well said

Prabhu

 John C. Chien September 23, 1998 17:34

Re: eddie &vortex, does true randomness exist?

Is it possible to design a long-chain molecule such that its scale is greater than the K-micro-scale . Can turbulence exist in the flow with very, very long-chain molecules ? I remember that Van Driest was interested in this subject many years ago as a method to reduce the drag.

 Jonas Larsson September 23, 1998 17:45

Re: eddie &vortex, does true randomness exist?

"TRUE CAUSE" is religion. Physics is model building based on observations. The Navier-Stokes model of a fluid is sufficient to explain all properties of turbulence, except maybe the pathological question of what triggers it (we know that something will anyway). In all real world cases quantum effects are negligeble, and discussing about them in relation to turbulence is like discussing if it is the "small settling dust particle", "the small vibration from a nearby freeway" or the "molecular vibrations" which makes one perfect sphere roll off another perfect sphere when placed exactly on top of it. The cause of the "roll off" is the unstable position, not the dust particle or whatever. In turbulence the "unstable position" can be fully explained by Navier-Stokes.

 R.D.Prabhu September 23, 1998 23:45

Re: eddie &vortex, does true randomness exist?

Agreed. one hundred percent!

 Jonas Larsson September 24, 1998 11:21

Re: eddie &vortex, does true randomness exist?

Cool idea, I also have a vague memory of reading something along these lines. I think it was quite recently, but I can't remember the exact application now. Should be possible to use long polymers I guess. I sure hope this doesn't become a serious application - would be a nightmare for us CFD guys ;-)

 R.D.Prabhu September 24, 1998 19:15

Re: eddie &vortex, does true randomness exist?

Starting from a laminar state (with just one mode k, say), can a turbulent field be produced ? To give it a historic touch this is what GI Taylor asks in his conversations with Batchelor. Well, I do not know if this question has been answered.

thanks

Prabhu

note: Dear Jonas, following our discussion you many note that the above question has been left out. And the answer to this is NOT ..the common one viz..turbulence is an instability etc ..

 Jonas Larsson September 25, 1998 04:08

Re: eddie &vortex, does true randomness exist?

I assume that you are talking about the "exact" Navier-Stokes equations here. As I see it this is a mathematical question and it is not something that we can observe in nature. I don't know the answer. Hopefully someone else can shed some light on the current status of this.

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