Hi all,

I am wondering if anyone could explain exactly what is happening when you filter the N-S equations. Physically you have a filter or width x, so in physical space how are do the model know which velocities belong to large eddies and which belong to small? If there are no nodes inside the filter size, then you have no information regarding the flow and cannot therefore decide what to keep and what to filter out....

Basically, I am asking for what all books do not tell us....what it is that LES is doing from a physical point of view and how is that related to the filter etc. I don't like words like filtering and averaging as they are not all that clear.....I know, maybe the question is not the best, but I need to know!! Thanks.

> I don't like words like filtering and averaging as they are not all that clear.....

I am certainly not an expert on LES, although I spent one semester as a graduate student writing an LES code for an unsteady compressible flow problem, using Favre mass averaging. After I analyzed all my results and wrote my report, I came to the following conclusion - and I am sure there are people who will strongly disagree with me:

I think the bottom line to LES is that you are throwing information away that would otherwise blow up your computation. The trick is to average or 'filter', i.e. throw away, just enough information so that you can still get a 'good' solution, one that does not look too 'washed out'. This is where 'turbulence modeling' comes in, of which I am not a great advocate either. Pick a different turbulence model for your subgrid scales, and you will get a different answer - a guessing game in my opinion. What bothered me the most about LES is the amount of CPU time you have to spend on 'throwing away' information that took time to compute in the first place.

References for Filtering

Physical meaning of filtering :: Fig.1.3 & Fig 1.4 of book by Durbin & Reif. " Statistical Theory and Modeling for Turbulent Flows "

Application of filtering in LES :: Ch.12 of "Fluid Flow Phenomena, a numerical Toolkit" Edited by P. Orlandi.

************************************************** **************** Filtering is not a simple problem in LES. (Ref. "The basic equations for the large eddy simulation of turbulent flows in complex geometry, by S.Ghosal and P.Moin, JCP 118,24(1995) )

There are many topics to be solved to implement filtering technique in real LES calculations. The followings are as far as what I have realized.

Commutation error, cut-off wave number(explicit filtering) or the smallest mesh size(implicit filtering), inhomogeneous mesh structure.

Due to these ambiguous problems, LES may contain errors in itself. In complex problems, not only these filtering problems, but also there are other errors from temporal discretization, spatial discretization, numerical scheme, turbulence model(SGS model), ... etc.

Based on my experience(Turb. flow around sqaure prism), these errors seem to be reasonable and acceptable when I compared my results with experimental data.

Everything Jongdae has said is true. To that, I would like to add the following observations (in the context of comments made by Axel).

LES predictions are not that sensitive to type of model used in complex problems. The predictions are usually much closer to experiments than the predictions made using RANS approaches. Using a simple LES model (and a lot of extra run time), you can get reliable predictions in most cases. If you use a more complex model (Lagrangian dynamic filtering, selective structure function etc.), the predictions do get closer to experimental data but the differences between predictions made using different LES models are much smaller than the difference between LES and RANS predictions.

What this means is that, if I know nothing about a turbulent flow, I can use a LES approach and be assured that results are reasonable. I could not say the same for RANS approaches which seem to have a lot of adjustable constants (curvature corrections, helicity effects) and thus a lot of empiricism.

While there is some doubt whether LES can some day produce results as reliably in combustion as it does in fluid flow, it widely acknowledged that RANS will never work for combustion. I know there are a lot of RANS approaches (based on PDF, CMC, second moment closures) that have been developed over the years which have been calibrated to produce reasonable results in specific cases (RANS models in KIVA for IC engines, PDF models for gas turbine combustors (see the literature by Pope, Correa) etc.). However, if you take a new type of combustor, the predictions will not be as good.

The problem with LES has been that it is mostly done on structured meshes (although high stretching might create a problem on them as well). Unstructured LES has not matured yet but there is a lot of work being done. The problem, atleast what I think, is that it is difficult conceptually associate a (implicit) filter to an unstructured mesh. On a strutured mesh, we do not know what the implicit filter (which depends on the discretization in space and time and the numerical scheme) really is. The only thing we can say that the characteristic (implicit) filter width is proportional to the grid spacing. The only thing we need to know for LES is the filter width for which we can use the grid spacing (by absorbing the proportionality constant) into the model coefficient.

It is to be realized that grid spacing is not uniquely defined even for anisotropic cartesian grids. One can define it as RMS of grid spacings along the 3 directions, or take the cube root of the cell volume or use more complicated measures as suggested in a series of papers by Meneveau and Scotti. The "grid spacing" produced by all these measures are probably not very different if the anisotrophy is not very high. Grid anisotropy is usually high at the walls but here, the turbulence is not fully developed (viscous and log layers which are dominated by the so-called lambda-vortices) and the assumptions behind LES are not true anyway (hence the wall corrections). On an unstructured mesh, however, it is hard to define a scalar grid spacing at a given location. It therefore poses some problems. The simple solution that people have adopted is to compute the solution on an unstructured mesh and project it onto a structured mesh and treat that as your LES field (this is what I guess Axel keeps referring to). It may not be elegant but this approach was quite popular unless some one has come up with a better approach, unbeknownst to me.