Generalized Math. Form of Turbulence
In one of his posts below, Tom says as a reply to "Sci":
"... Actually there is no accepted "generalized mathematics" of turbulence only conjectures and oversimplifications.... You also need to remember that chaos is not turbulence..."  Opps. Tom, are you a scholar professor? There isn't a "generalized mathematical form of fluid flow turbulence? Sorry  what equations are you "trying" to solve when you study on a turbulent flow? NAVIERSTOKES (or simply F=MxA)... Do you NOT consider this is the "most" generalized form of turbulence mathematics? Well, then, you are far from classical mechanics. Anyways  I guess you meant somethings else? My point is that ANY ATTEMPT to simplify a general turbulence equations by dimension reduction will cause big big errors ... AND.. as you know all research centers, universities, scholars, professors, etc are playing with simplified equation forms of generalized form of turbulence mathematics. Therefore, all of these today scholars are on the wrong way. Pity them and their students. You all have spent your life on the wrong ways of your master scholars/professors. 
Re: Generalized Math. Form of Turbulence
Dear Nomad
I guess that the problem is wider than you suppose. The main problem  all our world is built on the heat transformations and Newton's physics. Even computations become more and more problem of rising of temperature and consuming of electricity than result of computations. Somewhere reckoned that the computational productivity and energy producing will restrict each other. It still worth from bussines's point of view but not future as a whole. So let's solve our generalized forms of turbulence mathematics and spend our life on the wrong ways till new energy producing and flight fundamentals will be discovered 
Re: Generalized Math. Form of Turbulence
"all of these today scholars are on the wrong way"
Check out some literature and you'll find hundreds of validations of turbulence models against real experimental data. The predictions (and the researchers producing them) can't be all that "wrong" when computation and experiment agree well.... 
Re: Generalized Math. Form of Turbulence
Aha. And then there's this embarrasing problem of being able to get decent simulation results either for a plane jet or for a round one (doesn't get much simpler than that!), but not both at the same time with the same model. Turbulence models are a (necessary) simplification of NS which is moreorless adequate for engineering purposes. No more and no less.
Incidentally, there is lots of experimental results and lots of turbulence models out there but in the last 10 years that I have been looking, there isn't a single RANS model that does a halfdecent job across the whole spectrum of applications. On the basis of this, I personally look turbulence modelling with a slight "oh, yeah" attitude and choose horses (models) for courses (flows) as best I can :) 
Re: Generalized Math. Form of Turbulence
Computation is bringing NOTHING to the oldest problem of classical mechanics; i.e. Newton Physics.
First, you need to check if your main equations (Newton Physics in general and NavierStokes for flows as a result of Newton Physics in particular) are correct or not. BUT you have been unable to check your main basic equations yet. You do not have an analitical solution to generalized fluid flow equations, you do not have a complete direct numerical solution to these equations yet, etc etc. And, taking any simplified version of generalized equations will lead big errors if you remember high nonlinearity of these equations and very big instabilities due to them. "... So let's solve our generalized forms of turbulence mathematics and spend our life on the wrong ways till new energy producing and flight fundamentals will be discovered..." Well, then enjoy your spending your life on the wrong road. But be careful about not breaking the glasses when your hard works are helping on the discovers of flights. Your flights are giving big damages to the nature and global warming has a faster accelaration due to your flights/planes with full of errors. The life is so that where ever you go to (even in wrong direction) you learn and discover new things. At the end, you see all you have discovered are wrong. 
Re: Generalized Math. Form of Turbulence
"... Check out some literature and you'll find hundreds of validations of turbulence models against real experimental data. The predictions (and the researchers producing them) can't be all that "wrong" when computation and experiment agree well.... "
 Well, computation models use main basic Newton Physics (or Navier Stokes) equations.. So, nothing new here if you aren't sure about Navier Stokes' forms. As for experiments; You probably know that Experimental Technology too is based on same/similar mathematical equations of classical mechanics. I mean these experimental technologies too are based on such classical mechanic equations. For ex., take hot wire anometry. Whats its main principle? Newton's cooling special formula which relates the temperature/heat to the velocity. So, experimental technology today isn't independent than the classical mechanic equations. So, testing computational results by experimental results is meaningless, absurd. 
Re: Generalized Math. Form of Turbulence
"chaos is not turbulence"
ummmmm. Turbulent flow exhibits many of the characteristics of chaotic systems: sensitive dependence on initial conditions structural similarity across many scales aperiodic space filling One test to determine if a system is chaotic is to determine the largest Lyapunov exponent. If it's positive then the system is inherently divergent. I wonder if anyone's actually done this test on say DNS data?? 
Re: Generalized Math. Form of Turbulence
My main point is that there is more to turbulence than the mathematical theory of chaos. In particular many of the theorems in dynamical systems don't carry over to the NavierStokes equations (except in one or two special cases; i.e. 2D with periodic boundary conditions in both directions). In threedimensions existence of solutions on an arbitrary time interval has still not been proved and this is a prerequisit for the application of inertial manifold theory (although it does not guarantee the existence of said manifold).
Another way to view this is to try to derive the amplitude evolution equation for a TollmienSchlitching wave in a boundary layer using rigourous dynamical systems theory  it can't be done with present (rigourous) theory. In your example of calculating the Lyapunov exponent are you sure that when you do this that you are not assuming something about the solution of the problem that has not been proved? A good example of why you should not equate chaos and turbulence is if you take and ABC flow which is a laminar periodic (in space and time) solution to the NavierStokes equations you will find that the particle path equations are chaotic. However the flow is definitely laminar. 
what's so wrong with classical mechanics?
So there's something wrong with classical mechanics?
The whole idea of modeling is to provide a means of predicting real world behavior in a practical manner. No, models are not universal, they are limited by their assumptions. However you cannot deny they are useful when applied within their constraints. Take a look at all the systems and components that have been designed, in part or whole, using CFD methods. How wrong can classical mechanics be when those systems perform the way the designers and engineers intended? Extending your arguments, a CFD solution is not valid unless no modeling is done. Okay.... Just try doing DNS for a high Reynolds number flow about a complete aircraft. Gee, that's practical. I may get a result by the time I retire. Or I can use a turbulence model and get results in a few hours that match wind tunnel data to within a few percent. Or is a few percent from reality "too much error" for you? 
Re: what's so wrong with classical mechanics?
"The scientist is interested in the right answer, the engineer in the best answer now." Golder, H.Q.
"All models are wrong  but some may be useful" G. E. Box The real art to modelling is gaining the necessary experience and confidence to know how innaccruate you can be but stil be useful. Not all CFD is applied to mission critical applications where the fluid beahviour is inherent in the designed functionality. 
Re: what's so wrong with classical mechanics?
I couldn't agree with you more. And with the quote of G.E. Box! I like that one!
I just don't believe Nomad's blanket statements about how "everything is so wrong" can have much validity when these supposedly erroneous ways have seen numerous successful applications and validations. 
Re: the problem is....
Although these equations are a simplified form of general Newtonian Mechanics, lets suppose these equations are generalized form of fluid flows. So, you have a generalized form of fluid flow called NavierStokes Eqns. BUT ... you are not sure if these equations REALLY represent the physical form of flows BECAUSE you have no complete/analytical solution to these full equations set yet and you know these equations are based on many assumptions assumed when building them first time.
Now, the problem is... You all are using these equations as "correct reference"  implying/meaning that you suppose they are representing the physics of flows. REALLY SO? You are unable to prove that. BUT you get some useful results???? I'm sorry  but this way of thinking or current science philosophy i.e. the philosophy "..doesn't matter we are on the correct or wrong way  we are obtaining some useful applications.." is totally absurd and a shame of science scholars of our day/century. Donkeys too will produce some useful things when they go on the wrong ways. On all wrong ways, you will obtain some useful applications  but the dirt behind your backs will show you are on the wrong way. Since we see the dirt (air pollution, climate changes, etc) behind the back of science scholars today we can say they all are on wrong ways; from basical generalized mathematical forms of flows to experimental technologies based on same mathematics to computational dynamics which use errorrous initial equations. Really, think twice or more before you spend your priceless life time on such absurd approaches of science scholars. 
Re: turbulence and chaos
how are you defining turbulence and chaos? your definitions are similar to the ones the scholars (seen in the literatures) define.?
Anyway  Chaos is a subgroup of Turbulence. Randomness is maximum in Turbulence. (lets not use critical Reynold nr. to seperate Turbulence in another category.) NavierStokes are accepted to represent whole fluid flow turbulent motions in all time and space dimensions. BUT really so? UNLESS they are solved nothing is definite. Maybe, whole Navier Stokes equations are symmetric flows in all sections/manifolds/whatever you say and maybe this will be seen one day when NS equations are completely solved analitically. One thing all fluid flow scholars aren't aware of is that even symmetries in stable forms are flowing toward a chaotic structures and then fully turbulent motions where our analitical tools stop working  and some fully turbulent motions in some local regions actually may have very symmetric/coherent forms which we can consider minimum disorder in the flow. So, with millions of parameters sensitive to millions of initial and boundary conditions in any random flow motion, you will not be able to say anything about the flow characterists at any deterministic euclidian space. Dimension reductions using local/point/discrete/etc symmetries will only disturb the real physics of flow if NS really!!! represents the flow physics. 
Re: turbulence and chaos
Jeez, we might all be living in a bottle, the earth might be flat and 2" in diameter for all we know. Lighten up, have a cold guiness:) they've designed those cans using cfd for better taste, try it, it works....

Re: turbulence and chaos
"lighten up, have a cold Guinness"
Now THAT'S my kind of scientific philosophy! (and my favorite topic in fluids!) :) 
Re: the problem is....
"is totally absurd and a shame of science scholars of our day/century"
O dear. We live in a world benefiting from the application of engineering. So often I've seen this academic mentality detaching itself from the real world. CFD should not be the preserve of the scholarly few. Don't try to keep hold of it. Focus instead on automating it to the extent where it can be useful for all. Or do you want to keep it yours? Do you not want others to benefit from the application of CFD to their real world engineering problems? Do you feel that insecure? $$$s make the world go round and $$$s will ensure that commercial CFD grows and grows. If a company is not growing it is dying. A good thing too. History is peppered with examples of revolution against a priviledged few.... 
Re: the problem is....
Nomad.
Until you are able to come up with the Nomad Equations that define the fluid flow exactly even when discretization etc errors are present, we are simply forced to follow the mainstream and stick with the NS Equations and turbulence models. Or perhaps it suffices that you simply show that the NS equations are wrong. Should be enough for a nice thesis. Oh, wait a minute... how would you do that??? I give up, I am simply not smart enough to do that. Looking forward to seeing your work published in a journal.  Dude3 
Re: Generalized Math. Form of Turbulence
Tom, tell you the truth I don't know eough about the appication of the largest Lyapunuv exponent and what assumptions are implicit.
I'm just wondering if there's enough commonality between chaos (nonlinear dynamics etc.) and turbulence for there to be some useful cross pollination of ideas to further our understanding, and application of our uderstanding, of turbulence. One of the London universities was looking at fractal dimension and its relation to rate of energy dissipation I think. I seem to remember a few years back the work by York, Grebogi and Ott in applying small perterbations to a nonlinear system to coerce the system to a predictable state (pushing a strange attractor to a single point or closed orbit). In theory it held great promise. Similar sorts of things have been done recently with surface mounted microactuators, in conjunction with some fancy feedback control systems, in trying to reduce flat plate turbulent shear stress. 
Re: Generalized Math. Form of Turbulence
There's been a long interplay between dynamical systems and fuid dynamics which historically has proved very useful (and I believe still is to this day). However a number of dynamical systems theorists, many of whom had studied no fluid mechanics at all, made rather excessive claims about what they could do with the theory;i.e. much of this involved writing down there favourite normal form an claiming that it solves a specific problem without demonstrating that the required bifurcation/change in stability had/could occur in the physical problem. Examples of the miss use of dynamical systems can be seen in any of its applications to boundary layer theory where, because of the open nature of the flow, many of the fundamental theorems are inapplicable as they stand. However in closed flows, such as TaylorCouette flow or Rayleigh Benard convection in a closed container, many of the ideas are useful  the Shilnikov mechanism for the bifurcation to chaos in the TaylorCouette experiment springs to mind.
In short I'd say that ideas from dynamical systems are useful but be aware that sometimes the result you want to use has not been proven for the type of system you are interested in, hope this makes some since, Tom. 
Re: turbulence and chaos
And while drinking that cool Guiness you can ponder on why the bubbles fall down the glass rather than up. Next time have a look its quite interesting!

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