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Friday, September 3, 2010

Turbulence in Wavenumber Space

As Joe has been talking about the CMB in wavenumber or spherical harmonic space, I thought I'd bring up another area where it makes more sense to talk is wavenumber space than physical space:  turbulence.  Richard Feynman famously called turbulence "the last great unsolved problem in classical physics."

There are several reasons turbulence had boggled the brightest minds in physics, math, and engineering for over a century.  Physically turbulence extends over many length scales - think of a waterfall for example.  All of the kinetic energy gained from the fall must go somewhere and it turns out that somewhere is heat (and sound, but mostly heat).  But to turn kinetic energy in a fluid like water into heat, one needs viscosity.  In a waterfall, viscosity is effective at dissipating heat through motions on the order of 1 micron.  So to understand the turbulence in a waterfall that is something like 10 meters high one needs to understand every micron of the way.  On top of that turbulence is chaotic (in the technical sense of the word), meaning that it is essentially random and unpredictable.  As an example, take this visualization of jet of fluid entering a super-sonic flow.

There are other reasons turbulence is a really hard problem, but it turns out that what we call the "range of scales" problem is where thinking in terms of sizes makes more sense than thinking in terms of physical position.  For you math-junkies out there, that means an integral transform to either Fourier space (for things in boxes) or spherical harmonic space (for spheres).  Either way, when you compute the amount of power at each size-scale in the flow, you get a plot that looks like this for the turbulent magnetic field in the solar wind:

...or this for water in tidal channels:

 ... or this for simulations of solar convection:
Here are three different materials, three different temperature and density regimes, and even a collision-less plasma just for fun.  All are doing different things on large scales (small wavenumbers) and the very smallest scales (large wavenumbers), but in between all of them show a fall-off proportional to wavenumber to the negative five-thirds power.  In fact it's nearly universal - energy cascades from large scales to small scales the same way in all turbulent flows.  So a process that is chaotic, random, and unimaginably complex in physical space is really very orderly in wavenumber space.

22 comments:

  1. Nick,

    That is really interesting that "energy cascades from large scales to small scales the same way in all turbulent flows." I'm assuming you will say we aren't 100% sure why.

    Bye the way, I enjoyed the simulation video. It is amazing what people are doing with shell scripts these days. :)

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  2. Interesting. So perhaps wave number space could be measured using a multichannel grid array of microphones hooked to a discrete fourier transform oscilloscope time averager (frequency vs time averaged amplitude for each oscilloscopic channel plotted to the points on a grid)?

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  3. quantum_flux,

    Now that would be a fun experiment: fill a room with a grid of microphones, start playing different sounds/music, and start taking 2D transforms.

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  4. Yeah, except that the pressure waves are linear when talking about sound waves, I would expect all the microphones to produce roughly the same Fast Fourier Transform. When discussing turbulence there out to be different magnitudes and frequencies hitting each microphone in the grid, perhaps with frequencies on a logarythmic scale if it spans a large enough range.

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  5. Quantum_Flux,

    Good point. As you can tell I am not an acoustics expert. :)

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  6. Actually, perhaps that isn't the best way to measure turbulence. I was just picturing what happens when somebody takes a video recording on a windy day (pfffff, shhhhh, pfffpffffff, shhhhh) or blows on a microphone, the main problem is that a microphone diaphragm doesn't measure a wide enough pressure range since sound doesn't make very much pressure. Maybe turbulence could better be done with sensitive air pressure gauges instead, pitot tubes with piezoelectric crystals perhaps.

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  7. Quantum,

    Actually the best way to measure the a turbulent velocity field is with lasers. If you have some sort of reflective material (e.g. certain types of smoke, very small plastic disks in water) you can shine an array of lasers through the material and measure Doppler shifts to get extremely accurate velocity measurements.

    I should also point out that these transforms are taken in space, not time, so measuring some quantity at one location over time doesn't produce this spectra. Also sound isn't turbulence. Turbulence is a bulk fluid motion, sound is a wave passing through the fluid.

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  8. "I'm assuming you will say we aren't 100% sure why."

    Actually the cascade in spectral space is fairly well understood. Perhaps I'll try to muster up a post explaining it, but a very hand-wavy explanation is given in the Wikipedia article on turbulence.

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  9. Yeah, but turbulence creates pressure differentials which can be measured as very low frequency sound.

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  10. Hi there, I just knew of this very nice blog. I noticed Joseph's smart comment on Hamza Tzortzis blog and that led me to here.

    I am interested to know the readers views on the Feynman quote "the last great unsolved problem in classical physics". I want to know what would you perceive as a solution to the problem of turbulence.

    Let me elaborate more, the consensus among fluid dynamicists is that turbulence is a continuum phenomenon not a molecular one (perhaps that's why Feynman said it's a problem in classical not statistical or quantum physics). And therefore, the Navier-Stokes equations represent all the physics of turbulence. Now of course we don't have a general exact closed form solution to the NS equations. But with Direct Numerical Simulation DNS on super computers we are able to resolve this wide range of scales numerically using very fine mesh and small time steps. Now my question is, when people say "a solution to the problem of turbulence", would they be satisfied with a high resolution DNS ? or are they demanding a closed form theory that describes turbulence away from the Navier-Stokes equations ?

    These are thoughts that puzzled me for sometime and I am interested to have a discussion about them. Sorry for the long post.

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  11. I wonder what would happen if the 5 second average 3D dynamic and static pressures were measured for turbulent flow over a (hexagonal or orthogonal) grid array in a plane, and then data for each grid point plotted onto a bell curve of the 5 second average dynamic and static pressure increments vs the number of times each pressure increment was logged. That would provide a pressure distribution curve over time for each grid point in the plane as well as giving the mode and the mean pressures, perhaps being measured for a time period of 5 of 10 minutes to get really good data. That way people could begin to understand the pressure deviations over a geospatial cross section of turbulence. Perhaps that grid array of sensors could be slid back and forth to get a general picture of how the turbulence disperses through space as well.

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  12. Hossam,

    Thank you for your kind words and I will always try to answer in comments or posts any questions you have. Firstly, I believe you are correct about why ge says it is a classical problem.

    Unfortunately I am not enough of a fluid/turbulence expert to answer your questions as extensively as you may like. I would think a numerical solution with high enough resolution to explain and predict all the measurable effects would be sufficient to have "solved" the problem.

    I think Nick may be able to give you a better estimate how close scientists are to such numerical solutions, but as you can see from the simulation above, they can already do amazing things with codes.

    Also, I'm sure many physicists would like closed for solutions that also see to account for all measurable features. I word it in this way as sometimes we don't know the exact solution to a problem analytically but what we have is close enough to predict everything we see.

    For example, take the small angle approximation. As long as you swing a pendulum at very small angles the analytical solution where all sin(theta) are converted theta is "good enough" to account for everything measurable.

    Therefore, numerical solutions and "good enough" analytical solutions I'm sure would be sufficient.

    But Nick or others can correct me if I am wrong or can hopefully fill in more details as again, I am not an expert.

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  13. Thanks for your response, I was kind of brain storming rather than asking a direct question. I don't think there is an agreed upon answer for that (partly philosophical ?)question among the fluid dynamics community.

    I do have some four years experience with numerical models of turbulence and CFD in general. I am aware of the state of the art numerical techniques used to resolve turbulent eddies such as DNS and LES (like the video in the post). IMO, for almost all practical and engineering concerns, numerical techniques that are even less computationally expensive than DNS and LES are quite successful in giving 'good enough' predictions about the flow field. But I am asking from the theoretical point of view. Assuming that in the near future the DNS uncertainties are minimized and the computational expenses could be afforded in a greater number of universities (right now, only few institutions could afford such facilities), would that be THEORETICALLY sufficient for physicists ? I mean a great number of articles speak about the 'unsolved problem' without defining what exactly is the problem and what would be considered as a 'solution'. If the answer is 'YES this would be sufficient' then I can see its solution in the near future. but if the answer is " No, a high resolution DNS would show us the detailed flow history of turbulent fluctuations, but doesn't speak about other mysteries such as why does the flow goes into laminar to turbulent transition" Then I don't see a solution in the foreseen future. I believe the community would benefit much from trying to define what to expect as a solution to this problem.

    Again, I know there is no definite, agreed upon answer for this issue, but I would like to know your opinions on it

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  14. For me the problem would be solved. There are many problems that I believe can only be solved numerically and I am content with that. Solving realistic problems in nature is a very difficult beast.

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  15. I agree with you. Until someone proves that molecular dynamics are essential to the study of turbulence and therefore the N S equations are not adequate enough to represent it (there are speculations about this, but not an accepted theory yet), I would say that DNS is the solution to the 'Feynman' unsolved problem.

    By the way, can anyone cite the original source in which Feynman said this quotation ? I found this quotation circulated a lot but never found an original citation for it.

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  16. Hossam,

    If you consider a DNS of a system the solution for turbulence then we'll have to wait a while for solutions to things like the solar convection zone or even weather models. The dissipation scale for air is roughly a micron. If you want to simulate something like Kansas, you need roughly 10^33 grid points. It's going to be a very long time before we have a computer that can resolve 10^33 grid points.

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  17. I think to "solve" turbulence what you really need is some sort of mathematical tool to understand its effects without having to resolve every scale of motion (aka a perfect subgrid-scale model in a LES framework). I'm not sure what the solution would look like - I know a number of people have tried to create statistical models of turbulence without any real success. I think that this problem (and many others in complex, chaotic, highly-nonlinear systems) are going to require essentially new mathematics. Of course if I knew what those mathematics looked like I'd have a Nobel prize and a Field's Medal, which I most certainly do not, so this is all just speculation.

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  18. That's good insight to the problem. Yes it would be a long time before seeing any DNS for an astrophysical flow (unless some quantum computing ,which I practically know nothing about, pops out).

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  19. Just a quick question. Earlier up in the article, the author writes: "On top of that turbulence is chaotic (in the technical sense of the word), meaning that it is essentially random and unpredictable."

    I have only taken two courses in chaos theory and bifurcation theory (respectively) but the whole time, the professor kept re-emphasizing that chaos was unpredictable but not random. It is fully deterministic yet unpredictable -- this is the nature of the strange invariant set which becomes a strange attractor.

    If it were random and unpredictable, then would there be any difference between chaos theory and the theory of random walks and Brownian motions?

    This is a minor point but I just wanted to make sure that I had not been misinformed by my professor for the past year.

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  20. Vincent,

    You are right. Turbulence is indeed deterministic not random. Turbulence is represented by governing equations called the Navier-Stokes equations which are chaotic but fully deterministic.

    When fluid dynamicists use the word 'random' they usually mean 'irregular' not not that it's not deterministic. Perhaps this is because turbulence has been studied long before the science of chaos was established, so the word just stuck with them.

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  21. Thank you Nick for the very beautiful article.

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