UPD (29/08/2018): Is the RANS/LES way of turbulence modeling a converging process, i.e. will it hypothetically converge to a hypothetical big but single PDE system? Or will it inevitably disassemble into several non-overlapping PDE systems/approaches each of which is valid only for a certain turbulent regime?
After more than 5 decades of active search, it seems that the hierarchy of moment equations for the Reynolds stress and higher moments derived from the Navier-Stokes equations does not have a universal closure that could be applied to more or less wide range of flows. Numerous closures have been proposed since then. All of them contain heuristic arguments which are usually changed from case to case. So, does it mean that our understanding of turbulence will never be complete? Or do people still believe that a “magic closure” exists which can explain everything?
Related to your query, I suggest you follow http://web.stanford.edu/class/me469b/handouts/turbulence.pdf
https://www.youtube.com/watch?v=SgC4VJfZLsc
https://www.youtube.com/watch?v=GjWWtSrt-Ao
http://web.engr.uky.edu/~acfd/lctr-notes634.pdf
Yes, Direct Numerical Simulation , in other words we solve the full NS equations on extremely fine grids and for very small time steps , for a sufficient time for the variables to reach their statistical values. Unfortunately because of the required excessive CPU time we cannot do it for high Reynolds numbers as yet (around 800) . In 10 years time with the progress of computers we will be solving at much higher RE numbers. At the moment LES is the recommended procedure converging to a single PDE system (see also Argyropoulos and Markatos, Applied Mathematical Modelling ,2015)
Richard Feynman described turbulence as “the most important unsolved problem of classical physics.'' While working on visualizations and simulations of turbulent flows, we thought, "Why not ‘gamify’ a key geometric aspect of this complicated problem and ask players around the world to help us?”
for more information see
https://www.scienceathome.org/games/turbulence/math-behind-turbulence/
Dear N.C. Markatos
I take it with a grain of salt if even direct numerical simulation would be able to simulate turbulence flawlessly. The reason behind that is we always limit the model in a simulation to a confined domain by considering boundary conditions, as a result, we are somehow ignoring the correlation between fluid particle inside and outside of the considered domain, so for example, the transportation and dissipation of energy or heat would not be as accurate as reality. Secondly, mathematical formulation always tied up with residuals and numerical errors such as truncation error to this end, even though we can extend the computation to the very very high orders, the pure reality could not be simulated perfectly. It is a very profound argument not only in physics but also in philosophy as far as I know that stems from the definition of infinity. More accurately, we cannot simulate anything in the world as perfectly as God engineered. But in the industrial context, I am on the same page with you.
Regards,
Dear Hamed,
You may not have a confined domain, so we simulate for example the atmospheric boundary layer. Now if you have accurate measurements somewhere to use as boundary conditions and use DNS on sufficiently fine grid with sufficiently short time step you will get accurate solution EXCEPT for truncation error which you can maintain low and indeed you can quantify.This is much more than an engineer needs.
Best Regards,
Nikos
Dear N.C. Markatos
Completely agree with you in the computational context, not for the real-world scenarios. What I mean is somehow deal with the butterfly effect which implies even the farthest smallest decimals may have a role to play in the holistic simulation of turbulence like in nature.
Regards,
Hamed
It depends on the definition of turbulence, but my answer is "yes". For example, the Navier-Stokes equations, can have turbulent solutions, so therefore that form of turbulence is described. But the description doesn't do us much good. It's like saying the logistic mapping can describe a form of chaos. Yes, it does. But that doesn't tell us very much, in and of itself. You can describe something well, perhaps even perfectly, without saying much about it in a predictive sense.
Satistical solutions theory is one way to describe it, at least partially. See
http://www.numdam.org/article/RSMUP_1972__48__219_0.pdf
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