# How the value of Cs is chosen LES?

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 June 17, 2002, 10:24 How the value of Cs is chosen LES? #1 senthil Guest   Posts: n/a hello all After seeing papers on smagorinsky's model I wonder how they are chosing the value of the constant, Cs. Can anyone explain how it is chosen and what is affect on increasing or decreasing it. Thanx in advance senthil

 June 17, 2002, 14:21 Re: How the value of Cs is chosen LES? #2 Jongdae Kim Guest   Posts: n/a In the analysis of turbulent flow around square cylinder, Cs=0.1~0.2 is used ( I'm not sure why.). Cs(Smagorinsky constant) is a kind of incrasing or decreasing factor for the length scale. In LES, length scale of the computational element (or cell) is very important. We should use very small cells to analyze small scale turbulent structures. It's very ambiguous, isn't it? In my case, usually I use Cs=0.13 and sometimes I use Cs=0.1. Actually I have experience of Cs dependency analysis but I didn't publish the results. Because the trend of SGS modeling of LES is using the dynamic models. If you consider it is worth publishing my results, I'll consider it. Here I'd like to explain the rough results. If you use higher Cs, the velocity time history seems to be smoothed or filtered with larger size of filter (It is true. You can find from the filtered N-S equation using Smagorinsky model). In the case of small Cs, time history of the flow properties seems to be more turbulent. Now we can consider Cs as a multiplication factor to the numerical viscosity which comes from the modeling of SGS Reynolds stress. Larger viscosity (or damping) means that you can get results more easily and sometimes computational time also decreases. Actually it's not easy to choose the most optimal value of Cs. But if you use reasonable value (this is also unknow at first), you could get reasonable results which depend on your purpose. Jongdae Kim.

 June 18, 2002, 00:18 Re: How the value of Cs is chosen LES? #3 Ajay S. Parihar Guest   Posts: n/a Hi Senthil How are you? Its not any hard and fast rule while choosing Cs in Smagorinsky's model. You can vary it 0.2 to 1.2. But Kim is right that if you take higher value of Cs then it means you are tending towards higher numerical viscosity and and hence large scale flow field. In this case you will get results easily. But in case of lower value sometime solution starts diverging..so you have to choose optimal value of Cs for your problem.

 June 18, 2002, 04:22 Re: How the value of Cs is chosen LES? #4 senthil Guest   Posts: n/a hi ajay. iam fine. y can't u mail me. my id is here. Thank you guys for your replies. senthil

 June 20, 2002, 08:21 Re: How the value of Cs is chosen LES? #5 Rajani Kumar Akula Guest   Posts: n/a Hi senthil, if u want to use a proper value for the Cs, first u should have a proper idea of the problem u r dealing and the code. if u r using upwind scheme then u may not require smagorinsky model itself or u require Cs very small. remember that upwind schems are too dissipative. usually Cs will wary between 0.1 (by Deardroff) to 0.2 (by Clark). ofcourse these values are found by using experimental data .but remember most of these results are found from the simple cases. i think u should consider dynamic models. atleast u can solve the Cs value problem. but u still have some other disadvantages associated with the smagorinsky model. bye rajani

 June 22, 2002, 04:56 Re: How the value of Cs is chosen LES? #6 senthil Guest   Posts: n/a thank you rajani senthil

 June 25, 2002, 16:00 Re: How the value of Cs is chosen LES? #8 Mayank Tyagi Guest   Posts: n/a Hi Kalyan, About determining Cs in complex geomtries and flows, the various "cures" for large local variations such as lagrangian averaging, localized dynamic procedure (relaxation scheme for Cs) and or some sort of averaging (test filtering) are just mathematical arguements to use a "relatively smoother" Cs (Spatial and/or temporal variation) values. (Although some physical arguements can also be made). About test-filtering in complex situations: One can formally perform the filtering in computational space rather than physical space in case of curvilinear meshes (Proposed by Jordan). Vasilyev's group has developed filters on unstructured meshes too (some problems still remain!!!). About over-emphasis of scale-similar models, Isn't everything we do is overly emphasized (we all are saving the universe!!!). Scale-similarity model is nothing but the leading order term in the Taylor-series expansion of SGS tensor (parameter of expansion being filter-width). Similar results are obtained when applying de-convolution or arguing for tensor diffusivity model. Physically, it is representing the interactions of largest unresolved scales with smallest resolved scales and is completely expressible in terms of filtered or resolved fields and therefore should not be modeled (and must be taken into account explicitly!!!). Arguement being, The SGS model should do minimal modeling of the flow. However, scale-similarity should not be treated as a model by itself (it can't dissipate energy). Therefore, minimal level of SGS modeling is a mixed model with scale-similar as well as a dissipative part. About "smagorinsky fluids", It is interesting concept to look at from different view, probably no better than saying eddy viscosity is treated as fluid (material) property though we know that it is an outcome of a flow state (turbulence). However, if one realizes that numerical values of Cs in different part of the flows must be different due to fluid dynamical reasons, such an approach (i.e. constant Cs) must be abandoned in favor of dynamic procedure (despite it's problem of ill-conditioning and more...) Hopefully, these comments may help somebody. Mayank

 June 26, 2002, 14:06 Re: How the value of Cs is chosen LES? #10 Mayank Tyagi Guest   Posts: n/a Hi Kalyan, I must admit that you have provided really good examples where the "physical arguements" for scale-similarity must not be "physical". Also, I agree with you completely that not always you can give attach a physical explanation to a "mere" mathematical procedure. However, I was driving the point of being "mathematically" consistent at a minimal level. And there one lands up with scale-similar term (purely from mathematical arguements) as a leading term (later interpreted by Bardina's arguements for local interactions near the filter-cutoff ...again note that Bardina "derived" the model other way around...he had the "physical" arguement first and the model later!!!). Also, one must note that the need for the complete SGS model to "dissipate" energy will introduce some form of eddy-viscosity term (you can introduce all kinds of mathematical as well as physical flavors to this term too). Dynamic procedure can also be proven as a mathematical REQUIREMENT (ref. Oberlack). One might be tempted to attach all sorts of physical arguements to this "minimal mathematically" consistent model...after all we are solving "physical" phenomenon!! You've said it and I agree to it....Numerical issues...they are very important. Resolution may provide some insight provided it separates the numerical issues from modeling issues (it can be done ..ref. Geurts's work). It is often blamed on resolution if LES doesn't "work" as expected (it's the easiest reason to find...of course, finer resolutions are "expected" to reduce modeling as well as numerical errors...but they DON'T work this way). One must introduce a filter-width INDEPENDENT of grid-resolution (not many studies in literature addressing that!!!). Even I haven't addressed that issue in my simulations (and I'm aware of the effort needed to do this)...at MOST times, one validates the "predictions" against the data and a "fair match" arguement is sufficient enough to keep the Pandora's box closed. Therefore, my two cents on the issue of determining Cs: If we want a "general" LES methodology applicable to a wide spectrum of flows, the minimal requirements on SGS models yield a mixed model with dynamic evaluation of Cs. Strangely, I've written long responses though I seem to agree with most of what you've said. Take care. Mayank

 June 26, 2002, 16:26 Re: How the value of Cs is chosen LES? #11 Kalyan Guest   Posts: n/a It is precisely in the context of introducing a filter width independent of grid resolution, the work that Junseok seems to have done might be worth publishing. I totally agree with your comment about "fair match". But then again, not many people get funded to look at fundamental conceptual and theoretical aspects of LES. Such work is however needed to model combustion. I do not actively work on LES (or gas phase combustion) right now, I just try to keep myself updated. But of late, I have found work by people like Vasilyev (on wavelets), Kevlahan (using Gabor transforms which seems to supercede LES filtering) and McDonough (DDS subgrid models) very interesting and worth atleast keeping informed about. As some one with more than a superficial understanding about LES, you might finding their work (if you didn't already know about it) illuminating.

 June 27, 2002, 15:07 Re: How the value of Cs is chosen LES? #12 Mayank Tyagi Guest   Posts: n/a For a person not working actively on LES, you are impressive. Vasilyev's work on commutation errors is really nice (It provided a nice mathematical background for LES). One must note that these wavelet filters are going to violate the REALIZABILITY constraint (Ref. Vreman JFM). Therefore, we haven't found a satisfactory solution yet (If we stick to filtering!!!). Kevlahan's work on Coherent vortex simulation (CVS) looks like a great idea from SGS modeling perspective. The "gaussian" nature of such small scale (left after separating coherent vortices from turbulent fields) is easier for mathematical modeling. Problem is physically the small-scales can't be gaussian (ref. Tsinober, Warhaft, Sreenivasan). However, there is a lack of "good" approximation for such scales and one might be forced to prefer "simple and stupid" over "complex and correct". Mcdonough's work on DDS subgrid models. Just by itself, it looks like a great idea for generating synthetic turbulent fields and developing closure models (definitely more "physical" and computationally intensive than fractal interpolation (Scotti and Meneveau) and/or de-convolution (Adams, stolz), also ref. Domaradzski). Is SGS scale really a low-dimensional system?? I shouldn't make comments on that for I have read only couple of papers on it (though I feel tempted to make a few!!). I think most researchers are shifting from "actually" filtering the turbulent fields and then modelling the sub-filter scales. Conceptually it is okay, just gives too many problems mathematically and numerically. I can put some really exemplary works (most conceptually, their "effectiveness" is yet to be proven). Adrian's work on optimal LES using linear stochastic estimation (nice theoretically...I won't comment upon it's implementation for really complex industrial flows!!). TJR Hughes' work on multi-scale formulation (still needs work on SGS models though). Similar work from W-K Liu's group on RKPM formulation of LES. Pope's formulation of LES equations in terms of projection onto local basis functions (again needs work on SGS modeling). Layton and Galdi's work on function analytic foundation of LES theory (lots of promise there...need to show it on "real" problems)...and the list will grow into jibberish...so I'll stop here. take care Mayank

 June 28, 2002, 12:47 Re: How the value of Cs is chosen LES? #13 Kalyan Guest   Posts: n/a Dimensionality of small (subgrid) scales is indeed hard to get at. But fairly simple agents-based models have been developed for some very complex non-linear systems (some co-workers of mine actually work on non-linear interactions between bubbles using this method). So, the dimensionality issue doesn't bother me as much. However, the lack of any representation for triadic interaction in the DDS bothers me a little bit. How do scales exchange energy in this system. In a brief conversation that I had with McDonough, he did not seem bothered by it. He asked me to read a couple of his papers that I have not yet had time to. I have not quite look at CVS yet. But what interests me about Kevlahan's work is the use of Gabor transform (which loosely speaking is a localized FT). I do not know how or whether this fits in with Pope's projection that you were talking about (since I have not seen Pope's work on this yet). The ability to have local fourier bases might provide an opportunity to use closure (2-point) theories for subgrid modeling. Finally, a question. Is the lack of realizability in wavelets due to fact that the filters are not always positive filters or is there a deeper reason.

 June 28, 2002, 16:09 Re: How the value of Cs is chosen LES? #14 Mayank Tyagi Guest   Posts: n/a Formal representation of triadic interactions between subgrid and resolved scale should yield a plateau-peak form of eddy viscosity (the cusp/peak near the cut-off has been explained by Kraichnan and lot of other researchers and this behavior is very robust i.e. it doesn't depend strongly on the type of closure used for sub-grid scales). Subgrid-subgrid interactions do govern the dynamics of SGS and must be modeled. I am not sure how DDS will capture this behavior either (But I could buy the arguement that a logistic map or low-dimensional representation of N-S equations will yield similar behavior of eddy-viscosity). Pope's projection on local basis function is using splines (It is a very clever way to avoid filtering and other numerical accuracy issues, LNP 566, pp.239-265). He defines the resolved fields by a basis-function representation, so that the governing equations are ODEs for basis function coefficients. Realizability means positive filters and wavelets may violate that. Again, Vasilyev's representation of commutation errors in terms of moments of filter kernel tells that commuting filters must have vanishing moments. For symmetric filters, all the odd moments are zero, however for positive filters, one will always get a finite second moment (unless, the filter can be treated as a distribution function approaching Dirac delta i.e. no-filter). My assessment is that realizability and commutation-error free requirements are contradictory (I could be wrong!!!). Mayank

 June 28, 2002, 17:18 Re: How the value of Cs is chosen LES? #15 Kalyan Guest   Posts: n/a Triadic interactions between resolved and subgrid scales would mean disaster in terms of cost. Perhaps triadic interactions between smallest resolved scales and subgrid scales isn't too bad an idea. Based on my incomplete knowledge of DDS, I am not sure how the subgrid-subgrid scale interactions are accounted for properly without triadic interactions. I have encountered something like this done (and have used it myself a while ago) in the Linear-Eddy_Model (LEM) and one-dimenesional-turbulence (ODT). But these are not Eulerian models, they work with Lagrangian maps and I do not think DDS is one. Near equal but opposite cusps arise for forward and backward scatter at the cut-off. So an argument has been made that by removing the cusp, you are able to capture some near field backscatter. I had once looked at a paper on hyperviscosity (which if I remember right was based on the cusp) but obviously the fact that it is no longer used might render some credence to this argument. So, i would would be happy with a model for subgrid evolution. I say this primarily because subgrid-scale modeling would become more important in combustion unlike in fluid dynamic modeling (where, as you have pointed out earlier, something dumb and simple rather than complex and correct seems to work fine). By the way, what is LNP. I could not guess the name of this journal. What Pope does seems a lot like FEM or the spectral-element method (or B-splines). Anyway it is a good thing that some one like him is finally getting involved in LES. If only he can target engineers while writing his papers. I usualy take a long time before I start to understand his papers. Even then, I am not fully sure I understand them fully.

 June 29, 2002, 09:19 Re: How the value of Cs is chosen LES? #16 Mayank Tyagi Guest   Posts: n/a Lecture Notes in Physics (Springer-Verlag) (My mistake...I should have used proper acronym) What Pope does seems a lot like FEM or the spectral-element method (or B-splines). Yes it is similar, yet very different. FEM and spectral Element methods (current implementations) are still struggling with filtering issues on unstructured meshes. Since a field represented in terms of local basis function is a smooth field (in general!!), one would land up with ODES for coefficients in basis function expansion and NO FILTERING NEEDED. Anyway it is a good thing that some one like him is finally getting involved in LES. If only he can target engineers while writing his papers. Totally agree with you. Pope is a researcher I always look up to. I usualy take a long time before I start to understand his papers. Even then, I am not fully sure I understand them fully. Seems like I am not the only one....ha ha ha!!!

 July 3, 2002, 21:58 Re: How the value of Cs is chosen LES? #17 Jongdae Kim Guest   Posts: n/a Kalyan, I have finished postprocessing of the results (Only Grid Scale properties) on Cs dependency. Turbulent flow around two-dimensional square cylinder. Re=22,000 based on the cylinder section dimension and inflow velocity. Computational domain and number of cells are fixed. Cs are 0.1, 0.125, 0.15, 0.175 and 0.2 Unfortunately significant differences are not found in terms of time averaged flow fields. (Actually only time-averaged flow field and time history of drag and lift coefficients are available). In the case of drag and lift coefficients, small changes of fluctuating patterns are observed, which means that max., min., standard deviation values, Strouhal number are a little bit different according to the Cs values. But it's hard to find consistency based on the assumption that Cs is related to the filter size. Could you give me title of the paper that you suggested to read? I tried to find but ... Jongdae Kim

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