Hello all, Can LES equations provide DNS results in the limit that the mesh density satisfy DNS requirements?

Thanks,

CFDToy

Yes. If the cells become very fine , the filter will be also small (in the order of Kolmogorov scale). In this case, the results of LES will be a kind of DNS.

Generally speaking this is not true. The answer is yes "if your grid is your filter" like in Fluent.

I have yet to come across a code that does not use the grid as a filter.

Obviously not in commercial codes. But it was not specified what kind of code is used.

However, the explicit filtering is the main stream at the Stanford CTR.

Hi, If an implicit grid filter (cell vol) is used, I guess LES -> DNS is possible with turb K -> 0? What happens with explicit filters ..could it still be possible to derive DNS results?

Thanks !

CFDtoy

With grid filtering the equations effectively solved are the N.S. equations plus a term, the truncation error, plus another term, the commutation error (Which, in LES terminology, also contains the terms usually modeled with the turbulence model).

These two terms depend on the time step dt and the grid step dx. In the limit dt,dx - > 0 (with costant ratio dt/dx) the two terms go to zero while more and more of the turbulent motion is well resolved by the grid. When the grid spacing dx is, everywhere in the flow, equal to some fraction (it depends on the numerical scheme adopted) of the local kolmogorov scale then DNS can be considered reached.

With explicit filtering there is an external parameter, the filter cutoff lenght, which is somehow fixed. In this case the equations have to be effectively filtered (depending on the approach, the velocity computed at each time step or just the convective term are numerically filtered at each space location).

With this approach, when the grid spacing and time step goes to zero, only the truncation error goes to zero while the effective spectral resolution is fixed by the filter cutoff lenght which is fixed.

Considering that the resolution is fixed by the filter cutoff lenght, in the first case this is connected with the grid spacing so refining the grid you have better and better resolution until the DNS resolution is reached; in the second case the resolution is fixed and refining the grid the numerical error goes to zero. In the latter case there is grid convergence to an LES solution while in the former there is always, in the solution obtained, some percentage of numerical error until the resolution is such that a real DNS is performed.

When performing the explicit filtering, even if the grid is such that a DNS is performed, the part of frequency spectrum with frequency higher than that of the filter cutoff is always filtered out. Because the numerical error is mostly concentrated in the highest resolved frequencies, there will be a grid spacing for which the part of the spectrum contaminated by the numerical error will be filtered out and the grid convergence will be considered achieved.

In theory, to obtain DNS results with explicit filtering, the filter cutoff lenght should be connected somehow to the grid spacing, such that with dx -> 0 also the filter cutoff goes to zero. However, to obtain results consistent with the explicit filtering approach, the filter cutoff needs to be very big. For 2nd order central finite difference schemes the filter cutoff lenght has to be at least 6dx.

Anyway, DNS is not the target of a LES computation, so you will never try to reach it (also because you will probably never can because of limited computational power), doesn't matters the approach you are following.

The real difference between the two approachs is that, for fixed resolution, in the explicit filtering approach you pay some extra cost (usually a very big one) to completely remove the numerical error from your solution and to have a real LES solution (that is, a solution which satisfies the LES equations as they are usually written in all the books).

The grid filtering approach, while being more feasible for a fixed resolution, has several drawbacks. The biggest one is that if you are using a 2nd order scheme, the turbulence model will always be of the same order of the truncation error so...there is no theoretical reason to use it at all (except for the dynamic approach that has some nice features)!!

Probably i've been a lot confusing but i hope it helps