Hi Richard,

Rather than say that "is isotropy necessary to simplify the models", let me be a diplomatic by saying that isotropy is preferable to simplify the models wherever possible !! :-) :-)

But that apart, it is clearly a more valid assumption (than otherwise) for flow spots that are "away from boundaries". But then "close to boundaries" one may assume streamlined flow for quite a few problems.

Regarding your final question I would say that conservation of energy would generally take care of cases where some of the rotational kinetic energy gets converted to other forms "within conservation". Perhaps, that is the reason for its preference by analysts ?? Just a guess !

Cheers !

-Sanjay

Thanks Abhishek and Sanjoy,

Let me rephrase my question: Would models of turbulence that include anisotropy of angular momentum (a.m.) improve models in advanced engineering applications?

Example: models of turbulence are limiting factors in design of fusion reactors, according to an announcement of significant improvements in reactor design announced by a team at MIT last winter. 

Announcement: http://news.mit.edu/2016/heat-loss-fusion-reactors-0121

Journal articles: http://iopscience.iop.org/article/10.1088/0029-5515/56/1/014004/pdf

http://arxiv.org/abs/1505.03000v2

Improved modeling of transport of angular momentum (a.m.) at very tiny scales over several orders of magnitude enabled the team's models of turbulence to reproduce measured results. This was a key factor in making the design improvements. Their models enforce conservation of a.m. only globally; they cannot do this locally.

Dissipative processes certainly transport a.m. from orbital distance scales to small scales that are not modeled by fluid velocity. However, such processes cannot create or destroy a.m.; therefore they cannot destroy anisotropy. I suspect that including anisotropy at tiny scales is important. The math for doing this is not widely known (Riemann-Cartan geometry), so I suspect no one has explored this.

Thanks,

Hi Richard,

" Would models of turbulence that include anisotropy of angular momentum (a.m.) improve models in advanced engineering applications?"

Hmm.... :-) :-)

One of the apparently ageless classic questions that fluid mechanists, chemical engineers, mechanical engineers, and mathematicians engaged in nonlinear dynamics have wracked their heads on for decades !! It has generated huge amount of research culminating in volumes of publications, and led to the genesis of areas of work such as CFD development !!!

The simple answer to your question is perhaps continues to be (an ageless answer to an ageless question !!) "POSSIBLY, YES ! But will it be worth the effort ??"

And the answer to that counter question has depended on and continues to depend on:

• Do we understand the phenomenon (or should it be phenomena ??) of turbulence precisely enough to model it better ?
• Just how "messy" do our ODE's and PDE's get before we can solve them ?
•  Are our computers powerful enough to simulate the phenomena numerically ?
• Do we have sufficiently powerful numerical methods (hence software tools) to support us ?
• How hellish (and expensive !) will be the experiments as we go deeper and deeper into the damn thing ??
• At the end of it all, will we be convincing enough to the "crowd" ??
• And finally, will all this lead to meaningful improvement in science, medicine, or industry ??

So you-n-me are free to take our pick from the host of arguments and counter arguments !! Sorry if I made it all sound light-hearted and casual...

Why not try out anisotropic turbulence and see for ourselves ?? Will certainly look out for your findings and results...

Cheers, and GOOD LUCK...

-Sanjay

Richard,

isotropy in continuum models is always taken as LOCAL ISOTROPY. In my view it's a similar concept like LOCAL homogeneity or piecewise linearity (in non-linear relations). These concepts allow for describing anisotropic and inhomogeneous and even non-linear situations. The "pieces" need to be "small enough" - preferably infinitely small. See differential calculus - one of the basic tricks of continuum mech ...

I attach a paper where I applied these concepts to turbulence and got even real results like the prediction of the Karman constant as 1/sqrt(2 Pi) = 0.399. I used local homogeneity and local isotropy but no empirical data.

Article Universal equations and constants of turbulent motion

Richard, I recently applied the concepts described above to non-Newtonian fluids, with measurable success (in terms of industrial efficiency and thus $$$) applied in a factory in China. This was an application of my concepts. It made necessary to take additional phenomena into account whereas the basic idea (an integrated collection of earlier ideas of Landau, Kolmogorov, Hans Herrmann and myself) has not been changed a micron.

Such an application could solve the MIT issue of fusion. If a correct understanding up to dedicated control of turbulence is indeed a key issue then the colleagues should contact me for consultuations. However, this would not be for free as it will not be done in an afternoon. Here geometry matters but is surely no insurmountable barrier ...

Helmut, Would you be so kind as to send me a summary of your work? My objectives are first to understand how you altered the theory, and second to understand any impact on applications so far.  Does your work use affine torsion to represent intrinsic (subscale) angular momentum?

Thank you,

--Richard

email: [email protected]

Hi Helmut,

Thanks for the lead to your paper. I could get the final Physica Scripta 2013 version from arxiv.

Will be looking forward to your experimental results from China as and when you make them public.

In the meantime, I noted from your RG page that you are working on a wind tunnel turbulence project as well ! I will of course search the web, but have you published outcomes of this research anywhere yet ?

With best wishes.

-Sanjay

Sanjoy,  I agree there are some major challenges. After all, Feynman referred to turbulence as the most important unsolved problem in classical physics. In the past century mathematical theories were ahead of measurements in many cases (sometimes by decades): general relativity, cosmology,black holes, quantum field theory in many areas, the Higgs boson. Regarding anisotropic turbulence, I believe the critical limitation is knowing how to write down the theory.

I welcome these challenges.

Thanks, Richard

Absolutely and totally, Richard !

I would say way to go !!! After all, that is what research is all about..., isn't it ?

Cheers !

-Sanjay

Hi Sanjoy,

in the wind tunnel business there are similarly universal constants found like at solid walls the Karman constant. We want to compute them based on my theory. However, this is not too simple because at least my personal math capabilities are not sufficient to find an analytical solution or approximation; and I am afraid that if somebody finds one via certain series expansions etc. then the overall result is page-filling and nobody likes to know it because it is simply boring. One cannot sell it ...

(Compare with the analytical expression for Karman, 1/SQRT(2*pi), which is nice and may be printed on a T-shirt and sold for a good price, or burned on a mug and sold for a still better price...)

But seriously, before applying the theory to fusion, the windtunnel is a necessary intermediate step. Step by step, as our mothers told us ...

No, our results on windtunnels are not yet available. We even did not officially start as I still have to do some homework and look for funding to cover at least travel and other costs. If you have ideas how to overcome this bottleneck, you'ld be welcome in the team!

@Richard and Sanjoy:

I would agree with Feynman who described turbulence as kind of forgotten body in the fridge of Theoretical Physics. And it's Katepalli Sreenivasan who loves to repeat that turbulence is part of PHYSICS. And it's me who loves to repeat that Einstein was a 3rd class expert in the patent offfice in Bern/Switzerland when he came across with his first PHYSICAL ideas on Brownian motion and light. And it is me who loves to underline that special relativity is not more than the Pythagorean Theorem, if math is concerned. And it took A.E. many years to elaborate the math (in competition with Hilbert) to describe the physical idea of a generalized IDENTITY of heavy and inert masses based mainly on gedankenexperiments.

So in my irrelevant opinion, Theoretical Physics is more driven by gedankenexperiments and physical ideas then by math.

Math has matured from counting cows etc., from a scientific branch closely related with physical empirism, into an own science functioning in real as well as in virtual worlds.

Unfortunately, turbulence is part of the real world and one needs to know many empirical facts about it to formulate at least some right questions to pose, or: to formulate NOT the wrong questions. Even this is difficult because this branch has so many facettes.

However, a major but trivial facette has been neglected for a very long time: that turbulence is made up of vortices, ensembles of vortex dipoles. And exactly this was the key insight.

Many turbulence models do not assume isotropy of the larger scale (but still subgrid) turbulence. These models are broadly called Reynolds Stress Transport (RST) models.  There are many flavors of RST models.  However, you seem to be interested in large eddy simulation (since you used the word subgrid).  And RST are not often applied to LES. (But I have one on my personal website, Gadebusch paper).

The reason is cost/complexity.  Anisotropy tracking  requires computing a PDE for a tensor quantity (6 fully coupled unknowns).  Wall boundary conditions on anisotropy are difficult to implement (the Reynolds stress tensor appears at first glance to require too many boundary conditions at a wall).  Finally, anisotropy is not the whole issue for a better model.  The effects of fluctuating pressure are at least as important.   So RST models are less predictive than you might hope for all that extra effort.

In the cost versus predictive accuracy trade-off people routinely make poor decisions.  If turbulence represents 50% of the physics of your problem, then in theory it should take 50% of your computational cost.  But this is rarely the case.  Everyone hopes turbulence physics is trivial and can be short changed on the effort devoted to it.     Most people -  just want it to go away, and spend as little effort as they can get away with, on it.

Thanks, Blair.

You appear to have posed some serious realities that are in keeping with my light hearted earlier reply to Richard's original query.

Thanks for the brief on your wind tunnel activities, Helmut. I will look out for your publications and memos on that with interest.

Frankly, the reason I was attracted to Richard's initial posting happens to be my interest in wind turbines. It was only after entering the discussion that I realized his interest to be more in incompressible fluids than in compressible ones. Of course, as you correctly mentioned, the two classes of problems have similarities as well as strong differences.

If your "personal math capabilities are not sufficient to find an analytical solution or approximation...", then I must tell you that my own math is ZILCH !!

Cheers !

-Sanjay

@Blair: Gadebusch paper? I checked your site at ResearchGate but did not find something. Could you be more specific?

Second question: Your study "Determination of the decay exponent ..." (2011) is sort of internal report? It's an interesting and well written paper because the topic touches my own current interests, but I found some contradictory statements, e.g.in relation to Oberlack 2002 which I did not really understand. Due to my own theory the decay exponent n is unity as found by Oberlack (2002) using theory and by Dickey and Mellor (1980) using experiments with and without stratification (in the stratified case it's different physics, simple as mostly; I discussed that in a paper at my homepage here). Another point which was irritating for me is the notion of K/eps in your text. Do you come from empirical modelling a la k-epsilon by Launder+Spalding? That's not visible in the text but maybe problematic because these great engineers have had no problems to write down balance equations for non-conserved quantities like eps = dissipation rate. Auah, says the physicist, it's against the law...

Generally my impression is that you would nicely fit into our project about windtunnel characteristics, also Sanjoy. Besides, ZILCH = ???

Google "Blair Perot UMass".  Go to publications.

n=1  implies the turbulent Reynolds number will remain constant for all time.  No experiment shows this.  And it would be counter-intuitive to expect it.  After 100 years, you would expect the Re number to be smaller in decaying turbulence.  I would, anyway.

n=1 also implies perfect self-similar decay of the energy spectrum (Banard and Speziale ?).  But this is unlikely as there really are two scales in an energy spectrum (the peak energy and viscous scales).

Oberlack got n=1 for N.S. , but n = 6/5 for the Euler eqns using Lie Group analysis.  I don't understand any of if, and therefore don't trust it.  Sorry Martin...

I haven't written off  n=1 entirely.  But I think it is probably only possible at astoundingly high Reynolds numbers, or odd initial conditions (Vassilicos, at Imperial College)

The entire point of understanding turbulence is to model it.  So yes, my theoretical papers connect directly with turbulence model unknowns and terminology.  Dissipation rate is perfectly well defined, as is its PDE.  PDEs don't have to be conservative to be valid and useful.  The entropy transport PDE would be one of many examples.  You would be deceiving yourself, I believe, if you thought the terminology had any affect on the problem.

Thanks for the reply.

@ Blair: Thanks for the hint to Gadebusch, however download is only for cash. But the Abstract was already fine. You write "n=1 implies the turbulent Reynolds number will remain constant for all time".  Of course, in my theory Re = constant = \infty for all time. Maxwell also does not describe real electrodynamics but basic features like waves and their major properties. So please dont overstress theory.

You write: "Oberlack got n=1 for N.S. , but n = 6/5 for the Euler eqns using Lie Group analysis. I don't understand any of if, and therefore don't trust it." Well, so do I! The point is that NS is dissipative but Euler not. Or the other way around: NS is parabolic, Euler is hyperbolic such that for vanishing viscosity the character of the equations change qualitatively (think of the boundray conds!) If you wish to make Euler dissipative you need to apply a trick like 0*\infty = 1 at the high-wavenumber end of the spectrum. This issue is behind the two values for n although actually n should smoothly increase from 1 to 1.2 for decreasing viscosity. But it does not in the a.m. source - it jumps, so reflecting the qualitative change.

I remember great talks about that topic in a Prague pub with Feireisl who wrote together with Dafos a 3-volumes Handbook of Differential Equations, focusing also at this topic. Yes, sorry Martin, the last word on this actually classic topic has not yet been formulated.

You shouldn't write off n=1 completely because it has been measured by Dickey and Mellor in 1980. Serious people, serious experiments. Initial condition details do not matter at all in those experiments as long as the flow is turbulent and Re high enough.

You write that "PDEs don't have to be conservative to be valid and useful." Well, generally true, but specifically wrong, if you allow. A typical engineering point of view. We could talk about that if you once visit Hamburg or Berlin, we're sitting exactly between them (by autobahn and ICE only 60 to 90 minutes to those cities) and we have good pubs here, too. I already thought about organizing an Intl. Summer School on those topics here in the city, with hand-selected students and great young professors like you. Because we just had an Intl. Concurs on double-bass here in our castle, and we hosted master students of music from all over the world. It was a really nice experience.

Helmut, click on the title to get the free versions (some are preprints). 

All my papers are free on my website.

http://www.ecs.umass.edu/mie/tcfd/Publications.html

There is no experiment to date that I know of that has the accuracy sufficient to distinguish between n=1 and n=6/5 (or even 10/7).    It turns out that all exponents fit the data equally well.  Please try it.

If you don't want to bother, I have taken classic data from Batchelor and Corrsin (I thought their reputations could handle the criticism well enough). And shown that all exponents fit the data.  See.

J. B. Perot, Determination of the Decay Exponent in Mechanically Stirred Isotropic Turbulence, Advances in Physics, 1, 022104, 2011; http://dx.doi.org/10.1063/1.3582815

It turns out that everybody can find a data set that matches their theory because all data sets match all theories equally well.  One single experiment from the 80's is a very precarious handhold to rest a case on.   Try all the other reasonable possibilities (6/5 and 10/7)  and see that they work just as well.   Also note the tendency of humans to find what they are looking for (and n=1 was very popular in the 80s).  Still is really.  Won't die...  

I can get a DNS simulation to decay like n=1 for about an eddy turnover time or so (for certain initial conditions), but it never lasts for longer (and it should last forever). 

I can't prove that the Re is constant (or the spectrum fully self-similar with time) (n=0).  But I can do Saffman's proof that the Saffman invariant is a  constant (n = 6/5) if it exists, or  zero (n = 10/7).  

@Blair: Your 2011 paper on the decay exponent is drastically showing where the problem is: the experiments (including on computers) need to be dramatically improved. Not more but better work; conclusive experiments are urgently needed to differentiate between non-sensical and really predictive theories. Dickey-Mellor at MIT was not so bad in this sense because they did it with and without stratification and helped a lot to understand the interaction between turbulence and buoyancy forces in terms of resonance and energy conversion (from vortices to waves). But true, you have shown the weak point that it is difficult do exclude certain values in a log-log plot. I fully agree.

But still more: we need to look at other domains of knowledge outside the engineering world, e.g. geophysical disciplines like meteorology and oceanography where the Reynolds numbers are extremely high - even for free.

And we need to look at other approaches rather than sticking eternally to semi-empirical "modelling" with the many parameters, corrections, coefficients etc. which only cover our missing understanding.

Helmut, Sanjoy, Blair,

My main interests is in subscale a.m. and isotropy.

• Large-scale ordered KE becomes sub-scale disordered internal energy (IE) through dissipative (entropy-creating) processes. 
• In turbulence, large-scale ordered a.m. must become sub-scale a.m. Do any current models of turbulence contain a term that collects sub-scale a.m., and not just sub-scale internal energy?
• Although energy has no sense of spatial direction (it is timelike momentum in special relativity), a.m. does, so in complex situations a.m. is often anisotropic.
• I do not believe that anisotropy of a.m. can disappear through dissipative processes.

Does anyone have ideas or evidence contrary to any of the ideas above?

 From this line of reasoning, I conclude that:

• Any general theory of turbulence must have a repository for sub-scale a.m. that preserves anisotropy. This may be tough to deal with, but that is reality.
• It has long been known in classical continuum mechanics that the non-symmetric part of the momentum (stress) tensor is what feeds orbital a.m. to the sub-scale a.m." div(intrinsic a.m. tensor) = antisymmetric part of momentum tensor.
• Therefore at locations where orbital a.m. becomes sub-scale a.m., the momentum tensor must be non-symmetric.

Does anyone have ideas or evidence contrary to the line of reasoning above?

Sanjay: I have no preference for compressible or incompressible fluids, except that incompressible fluids are simpler to model, and the ideas I am pursuing are (I think) unaffected by compressibility.

Helmut: I believe that if we include the internal energy term and the viscous dissipation term, then the Navier Stokes equations with viscosity are conservation equations. Do I have this right?

Relation between theoretical math and numerical analysis

For hundreds of years, key advances in computational methods arise from finding better symbolic basis functions for particular kinds of models.

• Finite differences (whose basis functions are Taylor series evaluated at discrete points) enable solving many models in physics and engineering.
• Fourier analysis (whose basis functions are trig or exponential functions) help solve models with (almost) periodic solutions.
• Finite elements (whose symbolic bases consist of piecewise polynomial functions) enable computing models that involve complicated geometry, arbitrary spatial variation, and even singular (cracked) elements. 
• Wavelets (whose symbolic basis functions are products of trig functions and piecewise polynomials) enable solving for solutions that involve complicated geometry and arbitrary spatial variation with wavelike regularity.

All of these methods are instances of weak methods for solving ODEs and PDEs that have clear criteria (Sobolev norms) for converge of approximations to something real. This theory enabled quantum field theorists to work with singular solutions that are not ordinary functions.

Conclusion: a symbolic math problem would be very helpful in handling the enormous complexity of symbolic formulas that would result from a serious attempt to develop constructive methods based on anisotropic intrinsic a.m.

The relation between math and physics

I agree with Helmut's statement "we need to look at other domains of knowledge outside the engineering world".Mathematical insights are as important as empirical data.

• Maxwell's equations make sense in terms of Fermi-Walker transport and U(1) gauge theory. It is the ultimate classical (hbar->0) theory of electrodynamics (ED), not just a partial theory. Without the geometric insights, classical ED is a blizzard of curls and divergences and experimental data that can model capacitors etc.
• General relativity (GR) makes sense as special relativity + Newtonian gravity -> elevator experiment -> geodesic variation equation -> vacuum GR; plus conservation of momentum via the Bianchi identity -> full GR.These insights enabled A.E. to build a theory that was decades, even a century, ahead. By contrast, much that passes for theoretical research in gravitation today consists of adding ad-hoc terms to GR to see what happens.

The gauge theory view of ED and GR helped create SU(2) and SU(3) theories that are the foundation of the Standard Model in high energy physics.

A focus on empirical results reminds me of a joke. An intelligence test asked, "What is 2 + 2?" The mathematician said "4". The physicist when home and returned the next day with the answer "4.000000000". (He measured something because he doesn't believe in abstract theory.) The accountant answered, "What do you want it to be?" 

Thanks Richard, I enjoyed your contribution very much & have minor comments:

• >>Do any current models of turbulence contain a term that collects sub-scale a.m., and not just sub-scale internal energy?>>Yes, see attachment. In turbulence, angular momentum is discussed as vorticity.
• In turbulence we deal always with chaotically organized ensembles of vortices and thus only r.m.s. values are of interest. What you call "complex situations" is inhomogeneity of the driving forces of turbulent kinetic energy or rms vorticity. Locally they are isotropic, both.Why? Well, the vortices cannot appear as isolated individuals: like humans, they appear as couples called vortex dipoles. The have no a.m.
• Regardind NS eqns.: Yes, at least I would agree.
• "Conclusion: a symbolic math problem would be very helpful ... ": I agree again, but before discussing solution families we should most clearly describe problems to be solved, contradictions with observations or experimental results, and then one should concentrate at the most dramatic contradictions. And then: nothing is more successful than the success... I.e. state is free for everybody who has an idea.
• "Maxwell's equations make sense in terms of Fermi-Walker transport and U(1) gauge theory." Thank you, Sire, that M. did not spend all the work about ED and his life for nothing. More generally formulated: most of todays theory papers are what the notorious Wolfgan Pauli once called "Not even wrong", see the equally notorious Peter Woit here:

http://www.amazon.com/gp/product/0465092756

but see also the people's enemies here:

http://www.math.columbia.edu/~woit/TheCase.pdf

• "A focus on empirical results reminds me ...": Ok, good punch, hit midships. But we don't sink and insist that the kernel of theoretical physics is gedankenexperiment. And again: nothing's more successful than success...

Helmut

Article Universal equations and constants of turbulent motion

Hello Helmut,

It is a pleasure to converse with you.

• “In turbulence, angular momentum is discussed as vorticity.” Yes, I have seen that, as I have already been reading your paper. My thinking is fuzzy on this because I am trying to figure out whether a distribution of vorticities is needed at a location with non-isotropic vorticity.
• “they appear [only] as couples called vortex dipoles.” Is this assertion derived from the assumption of anisotropy? Is it possible that the dipoles are just the stable configurations that remain after the unstable states have made an important contribution to turbulence and disappeared?
• "Not even wrong": I agree that string theory became a tyranny that took almost all new tenured physics positions (in America at least) for more than a decade. This occurred because people when overboard over string theory’s ability to produce renormalized gravity. However, string theory has some things right: sources are not points, but lines; the strings have discrete lumps of curvature, (finite holonomy). Yet string theory took QFT as a dogma and that led it into 11 dimensions and sucked all the oxygen out of the room for less orthodox extensions of currently-accepted theory.
• “… we don't sink and insist that the kernel of theoretical physics is gedankenexperiment. And again: nothing's more successful than success…”: I believe that gedankenexperiment and intuition is the mother of creativity, not derivation from measured results. Einstein is the best-known example of this; in fact, no one would give young Einstein a recommendation for an academic position because he wasted his time on gedankenexperiments. According to Thomas Kuhn (Structure of Scientific Revolutions, 1962, 4th ed, 2012), the leading physicists of Europe adopted Newton’s law of gravitation long before the empirical data was able to definitively support Newton’s theory. I believe Einstein-Cartan theory (EC) is a third case in gravitational theory where intuition and math are already virtually conclusive, at a time when no direct experimental evidence is known.(Hehl and Blagojevich, 2013, and my own arxiv:1301.1588).

Bohr’s “old quantum theory” explained the discrete energy levels of atoms based on the flimsy conjecture that an integer number of wavelengths had to fit around an electron orbital. Alfred Wegener published his theory of plate tectonics in 1912, and was a berated for it (especially in America which focuses more than Europe on practical results) because no one at that time had a credible explanation for where to get enough force to move continental crusts, until the 1950s, a quarter century after Wegener’s death.

On the other hand, the ultimate case of justification because of computational success – Bohr’s Copenhagen Interpretation (that there is nothing else beneath the quantum wave function) – was the force behind a purge of all physicists who questioned it. John Bell described the tyranny he confronted from the keepers of the Copenhagen Interpretation.

Quantitative experimental proof seals the deal; usually it is not the source of insight. We need a balance; however, the balance is shifting in more toward mathematical modeling and intuition, because our models are now good enough in many areas (GR being the best example) that, in the right hands, breakthroughs often take place there first.

--Richard

Thanks, everyone.

That was a very interesting discussion indeed, and a goods set of shared publications as well. A very learning experience for me....

Cheers !

-Sanjay

All, here is the paper which Helmut Baumert mentioned earlier in this Q&A string:

https://www.researchgate.net/publication/303828981_On_turbulence_in_dilatant_dispersions

Article On turbulence in dilatant dispersions

Helmut, As you know I am not expert in turbulence; I am interest because of other fields. So please excuse me if I ask some dumb questions.

What turbulent phenomena depend on a non-Newtonian (decreasing) viscosity, as compared with a constant viscosity? Examples:

a) Formation of vorticity clusters (Bachelor couples in your papers)

b) "Low speed streaks", as described in Mccomb's 1992 book on fluid turbulence

c) bursting of streaks at higher strain rates

d) Reduction of flow resistance when certain filament materials (long polymers, fibers) are added to some fluids.

I am guessing that 

a) All these processes involve fluid "fracture" along surfaces in the fluid because the viscosity decreases sharply above some cutoff strain rate. By "fracture", I mean that the attractive forces that hold make it a viscous fluid are sharply reduced.

b) The fracture process requires increasing the potential energy at the fracture surface, because the molecules must be lifted out of the potential well formed by the potential wells of nearby molecules. This potential energy often comes from decreases in rotational kinetic energy of vortices.

c) Even though PE is increased by decreasing KE of vortices, angular momentum must be locally conserved in a truly correct physics model. Which models of turbulence incorporate local conservation of angular momentum and which do not?

Is this the right picture?

Thank you,

--Richard

Richard, I try to answer your question qualitatively:

In non-Newtonian liquids, the non-linear behaviour (and, as Helmut and I found, sudden chaotic behaviour above some critical shear stress which can cause bifurcation) is crucially determined by the inner structure of the liquid, which can be a dispersion or an emulsion (or also a suspension). The structure is built by the dispersed phase (usually nanoscopic), the droplets or particles arrange 3-dimensionally into pearl-chain-like filaments.

In our a.m. recently published paper, we are dealing with dilatant liquids, here, the structure becomes (for whatever reasons) denser and denser, and the higher the shear stress, the higher the viscosity, up to the extreme that the liquid even can become a solid!

You are referring to non-Newtonian liquids which show decreasing viscosity upon increased shear (if I understand you correctly), i.e. "pseudplastic liquids". Here, the original structure ("original" = at zero shear stress) gets destroyed, the particles are disaggregating, could form bigger (round) aggregates rather than pearl-chain-like filaments, in emulsions, the droplets can become bigger and would then lose contact (no 3-dimensional pearl-chain any more).

It is not yet possible to derive a general viscosity theory containing only structure and interfacial (energy) phenomena.

Richard, you said

"I am guessing that    All these processes involve fluid "fracture" along surfaces ..."

You would be guessing very wrong.  All of the turbulent phenomena you mention happen with constant viscosity.  I recommend the Turbulence  textbook by Pope.

Molecules are far too small to be relevant to the phenomena.

 There is almost no chance you can learn this extensive subject from a conversation or two.

J. Blair Perot, I don't think so, as here we are talking about non-Newtonian fluids.

J.B.P.,

You may be right that I need to read deeply. However, I have found in many field that if I get one or two insightful expositions, that is more valuable than a pile of textbooks.

You said: "Molecules are far too small to be relevant to the phenomena."

Last winter, the MIT Plasma group announced a significant improvement in certain fusion reactor designs. Improved models of turbulence that model  turbulence of ions and of electrons separately, -- with almost two orders of magnitude difference in scale -- enabled them to make this advance. I quote from the article at http://news.mit.edu/2016/heat-loss-fusion-reactors-0121: 

The expectation by physicists for more than a decade had been that turbulence associated with ions (atoms with an electric charge) was so much larger than turbulence caused by electrons — nearly two orders of magnitude smaller — that the latter would be completely smeared out by the much larger eddies. And even if the smaller eddies survived the larger-scale disruptions, the conventional thinking went, these electron-scale whirls would be so much smaller that their effects would be negligible.

The new findings show that this conventional wisdom was wrong on both counts. The two scales of turbulence do indeed coexist, the researchers found, and they interact with each other so strongly that it’s impossible to understand their effects without including both kinds in any simulations. …

Previously, scientists had thought that simply simulating turbulence separately at the two different size scales and adding the results together would give a close enough approximation, but they kept finding discrepancies between those predictions and the actual results seen in test reactors. The new multiscale simulation, Howard says, matches the real results much more accurately.

Richard, I could not find the MIT page about the fusion topic, possibly they have closed it due to security reasons. But what you are discussing is somewhat different from mechanics of turbulence. You talk about scalar turbulent mixing, which is governed additionally by the generalized Prandlt number. I know how to explain the phenomenon mentioned by the MIT colleagues because I easily could explain the mixing of heat. The whole thing is not trivial, but not too difficult once you have the solution for the mech problem. I would be happy to help the fusion colleagues to understand what they observe. I am quite sure they apply another collection of empirical rules etc. Possiby engineers in a domain of physicists. Oh oh oh...

Helmut,

Researchgate included the colon ":" in the URL for MIT News. The correct URL is:

http://news.mit.edu/2016/heat-loss-fusion-reactors-0121

Your article "Universal equations and constants of turbulent motion" is wonderful in several respects. First, on page 3, it discusses the role of intuition in science. Here is a link to Einstein’s wonderful comments on the subject:

http://www.neurohackers.com/index.php/fr/menu-top-neurotheque/68-cat-nh-spirituality/99-principles-of-research-by-albert-einstein

In section 2, page 3, you write, "... energy [for our particles] must be drawn from the mean flow field."

In section 2, page 4, you articulate an argument why the quasi-particles can have no a.m.: “If a particle would have its own circulation on the local level, it would add extra circulation to the flow, which would violate RANS. Our particles can therefore have no own circulation.”

I have three basic questions about current models of turbulence.

a) Is a.m. conserved locally? (The head of the MIT team, Prof. Ann White, told me that their model conserves a.m. globally, but not locally.)

b) Is there a field for intrinsic angular momentum (a place to store sub-scale disordered a.m.) that is analogous to internal energy (a place to store sub-scale disordered kinetic energy)?

c) Does the mean momentum tensor become non-symmetric in places where orbital a.m. is transformed to sub-scale a.m.?

 I expect the answer to all three questions (a,b,c) is “no”. In particular, (c) contradicts the dogma that momentum is always symmetric. I believe all the existing models of turbulence have it wrong on all three points (a,b,c).  Herein lies a great opportunity. (It has been known for a century in continuum mechanics that exchange of intrinsic and orbital a.m. requires that the momentum tensor be non-symmetric; in fact, the antisymmetric part of the momentum tensor equals the divergence of intrinsic a.m.)

The answer to all three questions is “yes” in Einstein-Cartan theory, which is a necessary extension of general relativity in order to conserve a.m. = orbital a.m. + intrinsic a.m. (which I have derived from GR in arxiv:1301.1588). EC is the only reasonable gauge theory of gravity that meets all current tests of GR, according to the experts Blagojevich and Hehl, in Gauge Theories of Gravitation, 2013.

 It looks like I may have to spend years to become expert in turbulence to prove this point.