A question for the DNS folks out there...

What kinds of length-scale are folks typically working with & what would be the smaller end of the element size range be?

Does anyone have any links to researchers in this field. I would love to see the intricate flow details.

Thanks so much.

diaw...

Kolmogorov's estimated in 1942 [ Kolmogorov, A. "The equations of turbulent motion in an incompressible fluid. Izv. sci USSR Phys, 6, 56-58 ] that is necessary a three dimentional grid with O(Re^9/4) to catch the smallest eddies of a turbulent flow through DNS. Since then, this relationship has been used to motivate the use of LES computations instead of DNS.

Do you have computer for it? ;o)

Cheers

Renato N. Elias High Performance Computing Center NACAD/COPPE/UFRJ Rio de Janeiro, Brasil http://www.nacad.ufrj.br/~rnelias

O(Re^9/4) grid POINTS

imagine an aerodynamic problem where the fluid has a viscosity about 10^6, thus you would require over 10^13 grid points to solve this problem with DNS.

Renato.

That really does seem like a huge number.

How would that translate into flow of air in a channel of height 1.5 mm, containing a flat plate inclined at angle 30 degrees to horizontal axis? Max flow velocity 10 m/s.

Re,Lp = (1.1614)*(10)*(1e-3)/(1.845e-5) = 629.48

(Re,Lp)^(9/4) = 1,984,802

Is that grid-points per 'what' dimension?

diaw...

------------- Renato wrote:

O(Re^9/4) grid POINTS

imagine an aerodynamic problem where the fluid has a viscosity about 10^6, thus you would require over 10^13 grid points to solve this problem with DNS.

Renato.

Good question!

The reference I have consulted doesn't specify nothing more about the Kolmogorov's law besides the relationship I've written in the last message. I think it's only a raw estimative.

Does anybody know more about the Kolmogorov's law that could help us?

Regards

Renato N. Elias

You need to resolve the smallest eddies, whose size is given by the Kolmogorov scale. Kolmogorov estimates that the ratio of the largest to the smallest length scales behaves as Re^(3/4), where the Reynolds number Re = u*L/nu is based on the largest length scale L (an appropriate length scale of your problem, like the length of a flat plate, the diameter of a nozzle, chord length of an airfoil...). The number of cells within a 3-dimensional L*L*L box would then be proportional to Re^(9/4). For low Re problems that's not to bad, but try a Reynolds number of 1e6 or higher!

Apart from this theory, it would be interesting to hear from a DNS expert what the common practice is. Do you really resolve the Kolmogorov scale?

For detailed chemical mechanisms involving things like hydrogen and methane at atmospheric pressures, the grid spacings are on the order of 5-10 microns while using high-order finite-difference methods. The rub is usually resolving the reaction intermediates that only exist within the flame.

Hi R_K,

Do you perhaps a copy of the equations simulated for these situations? What are the governing mechanisms that force such small grid-spacings?

diaw...

Actually the "order" estimate expresses the way the complexity grows with Reynolds number and does not include a coefficient which involves other parameters of the problem, and the coefficient can be quite large. The estimate refers to "degrees-of-freedom", not simply nodes. The number of nodes can be reduced by using higher order methods.

The turbulence cascade is believed to terminate at the molecular level, where each computational cell would contain only a few molecules and the continuum fluid hypothesis is no longer valid. At this point "fluid turbulence" would be indistinguishable from the random thermal motion, and one could say that the turbulent energy has been converted into heat. Since this scale is unobtainable for a macroscopic flow calculation, I think it would be safe to say that all DNS computations are really LES computations with very small filtering lengths.

Please critique this argument for me:

If the turbulence cascades down from larger to smaller scales in a way that is scale invariant, then it would seem that one cannot solve the turbulence problem computationally simply by mesh refinement, because no matter how small you made the cell, the fluid in it is still turbulent. At some point, could one do better by using higher order methods to try to better resolve the turbulence within the cell?

R. Lohner in his book asks the question, what conditions might be necessary to compute a fluid flow to arbitrary accuracy in finite time (which may be longer than any of us have on this earth)? With some (perhaps unreasonable) assumptions about the computational work per degree-of-freedom necessary to solve such problems, he concludes that the method must be at least second order in 2D and third order in 3D.

It would seem that higher order methods would be better for DNS computations, but the devil is in the details affecting the size of that coefficient mentioned at the start.

>The estimate refers to "degrees-of-freedom", not simply nodes. The number of nodes can be reduced by using higher order methods.

Yes, that's a good point. What methods are usually used for DNS: High order finite difference, spetral methods... ?

>I think it would be safe to say that all DNS computations are really LES computations with very small filtering lengths.

I see your point. But a problem is, if you want to consider turbulence all the way down to the molecular level, you will actually get in trouble with the most basic assumption of the Navier-Stokes equations: You cannot regard flow as an equilibrium continuum at that level. What do you think about direct simulation of the Boltzmann equation? Does it make sense to go that far,... and how well can we model molecular collisions.

I'm not sure you'll get much agreement on the "all DNS = highly refined LES" statement. If we are talking about phenomena that are below the continuum limit, then aren't we really talking about whether the NSE apply? I have heard that argument made, but the DNS computations that come to mind assume that the NSE apply and that there is some limit below which no resolution is required. On your argument: I am not convinced that it has been established that for an arbitrarily small cell we can still talk about a fluid or turbulence in the usual sense.

jason

I agree. That was also my point. I guess you have to make that distinction when talking about DNS. It's not really Direct Numerical Simulation of flow in the true sense. It's just Direct Numerical Simulation of continuum Navier-Stokes flow, with all it's assumptions. Is there a way to directly simulate flow? Even at the lowest level, you'll have to work with some kind of model, for example a molecular collision model, which is not based on first principles. Isn't that right? All we do is modelling, on one level or the other,... with a few fundamental principles thrown in.

http://www-personal.engin.umich.edu/...ymp05-0349.pdf

The spacing is driven by the sharp species profiles within the flame.

Thanks very much, R_K...

diaw...

I have re-listed two of your thoughts below, Mani. Both are very interesting as we now begin to touch on the lower level of applicability of the continuum assumption & its offspring, the N-S equations.

Is there a smooth theoretical bridge between the molecular mechanics approach & the continuum mechanics approach?

There is a fellow in Austria who performs very good molecular simulation work - give me a few hours & I'll find the links.

Dommelen does a lot of good work using a Lagrangian approach. I too have a lot of personal interest in taking this further in the long term. It may lend itself to providing a 'smooth bridge' between molecular & continuum.

Question: What is the separation dimension between molecular & continuum mechanics?

---------------------- Mani wrote: I see your point. But a problem is, if you want to consider turbulence all the way down to the molecular level, you will actually get in trouble with the most basic assumption of the Navier-Stokes equations: You cannot regard flow as an equilibrium continuum at that level. What do you think about direct simulation of the Boltzmann equation? Does it make sense to go that far,... and how well can we model molecular collisions.

Mani's previous post: I guess you have to make that distinction when talking about DNS. It's not really Direct Numerical Simulation of flow in the true sense. It's just Direct Numerical Simulation of continuum Navier-Stokes flow, with all it's assumptions. Is there a way to directly simulate flow? Even at the lowest level, you'll have to work with some kind of model, for example a molecular collision model, which is not based on first principles. Isn't that right? All we do is modelling, on one level or the other,... with a few fundamental principles thrown in.

Try this link to Vesely's homepage.

<http://homepage.univie.ac.at/Franz.Vesely/>

I find his work fascinating.

diaw...

I guess your "fellow in Australia" is Dr. G. Bird http://ourworld.compuserve.com/homepages/gabird/

BTW, I think the lowest scale at wich turbulence dissipates due to viscosity is not molecular level, but mean free path level, which is one order larger. And IMHO at this level NSE are still applicable.

Of course! I was just speaking loosely. But I think the continuum limit is probably unreachable computationally in a practical sense, and everyone has to stop mesh refinement somewhere. Just stopping without a model for smaller scales, is in itself a filtering of smaller details.

There is an article on turbulence in the September 2005 SIAM News by D. Holm and E. Titi that I found very interesting. Perhaps it would be more interesting if I understood the details. It gives an overview of turbulence and what one would like from a model, and talks about the LANS-alpha model (Lagrangian-averaged Navier-Stokes). This model truncates/filters the turbulence cascade at some scale alpha while preserving as many of the fluid properties, such as the circulation theorem, as they can. They claim to reduce the Re^(9/4) complexity growth to Re^(3/2) while preserving a bunch of fluid properties. Further work is being done as LANL and NCAR. This is a "descriptive" article, but a list of references is provided for more detail.

The book by Pope on turbulence explains quite well what is required for a good DNS. Briefly, the energy produced by forcing or a mean shear needs to be dissipated by the small turbulent scales. So these dissipative scales have to be resolved. Pope shows spectra of the kinetic energy dissipation in his book. You can observe that you catch most of the dissipative scales if dx = 3 * eta (eta is the Kolmogorov length scale). This is because the peak of the dissipation spectrum is not at eta but at about 10* eta. Most of the dissipation takes thus place at scales 10 times larger than the Kolmogorov length scale. If you do not resolve a large part of the dissipation spectrum you are in trouble, whatever the accuracy of your numerical scheme. The flow cannot drain its energy and the results are simply unphysical. An alternative is to use a low-order scheme (MILES) or a LES model to drain the energy.

The required resolution depends also what kind of statistics you want to compute. If you are interested in just the mean velocity it is not necessary the resolve the small scales very accurately, but if you interested in higher-order statistics (intermittency of the dissipation) a higher resolution might be necessary.

Another point. In DNS not only the resolution is important but also the domain size (should be larger than the largest scales), boundary/initial conditions and sampling time. Sometimes the errors caused by bc's, a too small domain or too few samples are (much) larger than the errors because of the finite resolution.

I hope this makes some points clear,

Tom

In principle, what you are saying is correct:

"Just stopping without a model for smaller scales, is in itself a filtering of smaller details."

However, the fundamental idea of a DNS is that there are no smaller (important) scales. Under-resolved simulations can probably be thought of as LES-like, and that can certainly be argued for older wall-bounded simulations that were called DNS's at the time. Tom's message in this thread talks about the dissipative scales, and I believe that that is the correct viewpoint.