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

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.

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.

hi ajay. iam fine. y can't u mail me. my id is here. Thank you guys for your replies. senthil

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

thank you rajani senthil

Jongdae,

I would most certainly be interested in seeing your observations regarding Cs published. Yes, the dynamic modeling seems to work well in LES and the idea of scale-similarity that is behind this approach is also physically appealing. However, its suggested use in all situations and some of the claims are overblown. This is true both from theoretical and practical view points.

From a practical stand point, averaging of Cs in the homogeneous direction is a problem. Usually, there is no homogeneous direction in most real flows of interest. Techniques that do not require averaging along homogeneous directions have been proposed but have not been established enough to guarantee their stability in all situations. The test-filtering also poses a big problem on non-cartesian (curvilinear or unstructured) meshes. In many situations involving chemical reactions, the fluid dynamic scale-similarity is violated in the presence of exothermic heat release (dilatation). This is especially true for flamelet combustion (both premixed and non-premixed) where reactions fronts are very thin. If you try to apply the test filter across the flame, you would end up with highly unrealistic values of Cs. Also the flow structure changes drastically across a flame front. Even if you had a homogeneous direction, averaging along it may be problematic if the flame front cuts across your averaging line. Some points on the line may be upstream of the flame (reactants) and other may be downstream of the flame (products). Since the turbulence on the products side looks a lot different from the reactants side, averaging does make any sense.

It would be nice to come up with an optimal value of Cs theoretically for a given flow and then use the standard Smagorinsky model and then compare it with the dynamic model. Sometimes, you get the feeling that dynamic model, for all its mathematical inconsistencies and ill-conditioning, some how hits upon the optimal value of Cs depending the flow structure. Yes, it is true that the scale-similarity is physically appealing but if you looked at the spatial variation of Cs values (unaveraged) even along the homogeneous direction, there is a very large variation. So, Cs depends greatly on the local flow structure which supposedly is the advantage of dynamic modeling. But by averaging, there is no such advantage, all local variations along homoegenous directions disappear. Had the Cs values not varied much along homogeneous direction when compared to its variations along other directions, then one can buy the argument.

Hence I feel that some insight about Cs is still needed in LES. There was a paper in JFM in the last couple of years with the words "Smagorinsky fluid" in the title. It presents an interesting view of Smagorinsky LES. It tries to study the (pseudo) dissipation range in a LES created by the Smagorinsky model based eddy-viscosity much the same way the kinematic viscosity creates the real dissipation range (in experiments and DNS). Cs in this paper is related to the shift in dissipation range from its actual location (at the Kolmogorov wavenumber) to the resolved wavenumbers in a LES. This is in tune with your suggestion that Cs value signifies the filter width and the associated shift in resolved wavenumbers. I suggest you read this paper (although I have some minor problems with some of their arguments that I shall not get into here) before you start writing your paper.

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

The idea of "representing the interactions of largest unresolved scales with smallest resolved scales" which is the corner stone of dynamic models is indeed elegant and I have used it often with good results for incompressible, constant density flows.

That said, have you tried to use these models in regions where scale similarity is suspect like modeling of turbulent flames. Also, not always and not at all scales are turbulent flows fully self-similar (e.g., turbulent flows dominated by strong vortex/conherent structures like the von-Karman, Kelvin-Helmholtz vortices). How you filter out the effect(s) of coherent flow components is not clear. The SSF model by virtue of its name seems to suggest that it does this very thing, but when you study the model, atleast I do not see how.

The scale-similarity is also being extended to scalar transport and dynamic models are being used for modeling eddy diffusivity and scalar dissipation. Scalar dissipation is a term dominated very much by the smallest of the scales. Scale-similarity between largest of unresolved scales and smallest of the resolved scales does not help since one needs information about scales that dissipate scalar energy (much like the need for SGS dissipation that you pointed out). Scalar dissipation unlike kinetic energy dissipation is less clearly understood because of the additional parameterization due to the introduction of diffusion time scale in the scalar spectrum (in addition to the viscous and inertial time scale that uniquely characterize the energy spectrum).

Finally, one need to be cautious in using smallest of resolved scales for anything since the high wavenumber spectrum is highly dependent of the numerical scheme (numerical dissipation mostly). If I end up with different values of Cs for the same problem when I use different schemes, what does that mean. If the results are still the same, then perhaps one can argue that use of dynamic modeling alleviates some of numerical artifacts. Also, there is no guarantee that scale-similarity in each of these simulations is identical to the scale-similarity in real turbulence. The dynamic filtering is then a mere mathematical trick and there is nothing necessarily physical. I bring in the discussion about numerical aspects since they are going to be relevant in practical applications of LES. The academic community seems to emphasize things like commutative filters, kinetic-energy preserving advection schemes which seem possible only for simple problems in an academic environment. I know that there have been attempts to extend these kinetic-energy preserving schemes for variable density, incompressible flows but without much success. So I suspect that all this hype about dynamic modeling in academia is a little self-serving.

To give you an example, I remember this paper out of UTRC which used various LES and RANS models for modeling non-reacting flow in wake stabilized combustors. If I remeber right, it was presented at the 97 Joint Propulsion meeting. The LES results (using various models) were marginally better than unsteady RANS models in terms of predictions. So they conclude by saying that better resolution is the only salvation (the grid they used wasn't bad either).

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

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.

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

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.

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

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.

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!!!

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