sorry, i mean second order central difference scheme for convective term

When you perform LES, you have to use a scheme with the less possible numerical dissipation, in order to insure against the risk that that the numerical (scheme) dissipation is lower than the LES dissipation (dissipation added to mimic the energy transfer to the unresolved scales). Then the scheme has to be at least second order, and if possible centered. This problem is less critical in URANS computations because turbulent viscosity is far greater that the LES subgrid scales viscosity. Another approach, known as MILES, is to use dissipative scheme (upwind) without subgrid modeling. The numerical dissipation is then assumed to mimic the dissipative behavior of the unresolved scales.

Hope this helps

Best regards ~ ~

Hi, Lionel,

Thanks for your help. But as we know, when the Peclet number is larger than 2, the second central difference scheme for the convective term will result in the numerical unstability, why does it work in LES? And another thing, in SGS model (e.g. dynamic model), the SGS stresses contain h**2 term which h is the grid size. This suggests the difference schemes for all terms in NS eq. should at least be third order O(h**3) in order not to mask the effect of SGS stresses. Is that right?

Thanks again.

Edward

Dear recipients.

I have been working on the simulation of a cyclone separator utilizing SGS models. I first started with an axisymmetric model (2D) and the Smagorinsky model. I have experimented with different convective schemes (upwind, central difference, power-law), but I have faced an interesting effect. THe finer the grid, the higher the swirling velocity, so that the turbulent viscosity is close to zero and the swirling speed is as twice as great the experimental data. Do you think this might be owing to the axysimmetric assumption? Thanks.

I don't knows if its the reason of such a strange behavior, but the Smagorinsky model (like most of the SGS models) is designed to be used with 3D flows. Usually, 2D LES doesn't mean anything because there is no possible energy transfer between scales in the third direction (unless you are studying 2D turbulence, for instance stratified flows). Another problem is that the standard Smagorinsky model behaves really bad when using with wall, and that it doesn't take any rotational effect into account.

Best regards

There is more to the scaling of SGS models with h than just the factor of h**2 in the SGS stresses. The factor of h**2 multiplies some other stuff, and you have to account for the scaling of the other stuff with h. This may vary slightly from one model to another depending on the closure approximations used, so you will have to work out the details for whatever model you are using.

It is unlikely that your problem is due to the assumption of axial symmetry, but two other things come to mind (besides the obvious suggestion to check your code for bugs). First, I have seen LES solutions apparently get worse with refined grids, and sometimes the resolution of the problem is that most experimental data has been averaged, while well-resolved LES solutions can include some big fluctuations. This is one aspect of the problem that for some flows, the ensemble averaged flows don't much resemble many of the individual elements of the ensemble, which is what LES produces. Second, 15-20 years ago, astrophysicists working on star formation discovered that strict angular momentum conservation was crucial in avoiding certain plausible but nonphysical numerical solutions. The solution to their problem was to use the strong conservation form of the angular momentum equation (that is, transport density * r * v_theta) and difference it with a conservative difference method.

It is generally agreed that the truncation error interferes with the Smagorinsky type LES model if the overall scheme is 2nd order accurate. If possible, a fourth order scheme may be preferrable. Fifth-order, upwind biased method is also an option although it has been contraversial from time to time.

Dean,

I agree with the 2 suggestions you have made but I would not rule out the axisymmetric assumption as a culprit.

I am not sure what turbulence in fully axi-symmetric flows looks like but 2D turbulence differs significantly from 3D turbulence in terms of energy transfer, reverse cascades etc. Also, if you assume axi-symmetry, center-line behavior is suspect since the radial component of velocity is always zero. The mean radial velocity may be zero (if the mean flow is axi-symmetric) but the fluctuations are not. If you use a 2D, axisymmetric code, even the fluctuating component of circumferencial velocity is also zero. I am not sure how one can predict the flow behavior at the centerline under such circumstances.

Even if a LES model works well in shear or homogeneous flows (for which LES models are typically tested), as pointed out earlier in this forum, there are no guarantees when it comes to rotating flows. ICASE has done a lot of work on rotating turbulence and it wouldn't hurt to take a look.

Kalyan,

I agree with your comments on the conventional wisdom being that 2nd order methods and the Smagorinsky LES model not being a great combination, but my point was rather different: One can't look at just the power of h when figuring out the scaling of truncation errors of the cfd scheme and SGS stresses in the turbulence model. You have to consider also what those powers of h multiply. A numerical method that is formally fourth order is not necessarily accurate on any given problem (trivial example: an unstable fourth order method gives wretched accuracy). Another aspect of this same issue is that you almost have to consider specific combinations of numerical method and turbulence model. One cannot make general statements about the adequacy of methods of a certain order or class (e.g., upwinded) based on the rather limited number of cases found in the literature. The trends are useful, and they should be heeded, but the prudent researcher will do his/her own analysis of the particular code and LES model actually being used.

Dean,

So, what do you think about 2rd central difference working even though pelect number is larger than 2.

thanks.

Edward

What do you mean by the phrase "it works?"

What evidence do you have that it gets an accurate answer?

I'm not trying to be evasive, I just need more information about what prompted your question.

In any case, I never use second order centered differencing for the convection terms. Almost all of my work is done with explicit transient codes for which this difference method has unacceptable stability problems.

Your comments are indeed pertinent, but this behavior has also been observed if the dynamic sgs model is used (which does not need to be tuned by means of a constant). Actually, we believe that the lack of a third coordinate is responsible for that abnormal behavior, even though it was possible to obtain good results in our 2D model by tuning the Smagorinsky constant and limiting the grid refinement. Obviously, we are regulating the effect of the unresolved scales. I have already posted this question in this forum, but I haven´t gotten a convincing answer. Numerically speaking, what is the difference between LES and RANS if the same pv coupling and the same numerical schemes are employed? I mean: the conceptual difference is obvious (resolved and unresolved scales in LES and mean velocities and fluctuations in RANS), but within the algorithm, the only difference is how the turbulent viscosity is computed. Thanks.

Hi,Dean,

I used 2rd central difference scheme for convective terms to simulate the channel flow and open channel flow. The results are quite well with the comparison of DNS results from stanford university. I just cannot figure out why this scheme can work for these flows. If you look at the website in CTR at stanford university, you will find that almost all of their LES work are based on 2rd central difference. You said you got unacceptable results with this scheme. could you please tell me what kind of problem you simulated. Thank you very much for your help.

Best regards,

Edward

Here are two papers that deals with numerical discretization issues in the context of LES.

Rai and Moin, JCP (1987 or 1989) Mittal and Moin AIAA Journal (1997-2000)

I do not have the precise dates when these were published but I do not think it is too difficult to find them. Both of these papers deal with accuracy of 2nd order central schemes when compared to higher order upwind/upwind-biased schemes.

> Numerically speaking, what is the difference between LES and RANS if the same pv coupling and the same numerical schemes are employed? I mean: the conceptual difference is obvious (resolved and unresolved scales in LES and mean velocities and fluctuations in RANS), but within the algorithm, the only difference is how the turbulent viscosity is computed. Thanks.

But that is the point of turbulence modeling isn't it? Afterall you are trying to solve the _same_ set of equations (N-S equations) supplemented with a model. How turbulence is modeled is the issue.

Even if we just look at LES alone (and avoid the discussion of the differences between RANS and LES), and focus for now only on the Smagorinsky _model_, there are many many different ways of obtaining the constant of the model, and that alone makes or breaks a simulation (even though the rest of the LES model is "identical" in all of these methods). I hope I'm being clear here.

One should note that the underlying assumption of Smag. models is the concept of turbulent "diffusion". This is not necessarily the correct assumption (in fact it isn't by all accounts) So if you wish to see a fundamental difference then you have to look at turbulence from a fresh perspective. In that case, there may very well be a big difference between RANS and LES (even at the "cosmetic" level)

Adrin Gharakhani

> the SGS stresses contain h**2 term which h is the grid size. This suggests the difference schemes for all terms in NS eq. should at least be third order O(h**3) in order not to mask the effect of SGS stresses. Is that right?

I'm not sure if I can agree with the logic (and much of the discussion on this thread) that the solution method has to be O(h^3) to avoid "masking" the SGS effects which contain a h^2 term.

There is clearly confusion (and that is quite common) in understanding what O(.) means. O(.) stands for order of magnitude. It gives absolutely no information about the actual magnitude of the error or the solution. All it says is that as you reduce the grid size by h, the truncation error will be reduced by h^3. Period.

Your model accuracy, which contains an h^2 term has nothing (very little) to do with the order of accuracy of your numerical scheme. Let me explain what I mean before people disagree. Suppose that you have an infinite order method, and let's go one step further, suppose you have exact info on the resolved flow field (no trunction error). Your model is still as good as C*h^2*|S|. So now as you take larger h's your turbulent diffusion increases. This is not necessarily correct. That's why there are dynamic models (and other sophisticated approaches).

Another point regarding the confusion with O(.). Just because you have a O(h^3) method it doesn't mean that you have a more accurate solution than some other O(h^2) method. For a given h, it is quite possible that depending on the discretization scheme you can have a more accurate lower-order scheme. All the order says is that your solution is guaranteed to converge faster for the higher order method (so in the worst case scenario, there will be a cross-over point where the higher order method is actually more accurate too)

Adrin Gharakhani

Adrin,

Your explanation of what order of accuracy means is right but some of the points made here are quite relevant.

Accuracy of LES often has as much to do with numerical discretization as the model itself. Sometimes, the LES models (even the so called dynamic ones) and the discretization errors are so closely coupled, it is often difficult to separate out the effects of each one of them.

LES is not a field theory. Almost everywhere, it is explained as if the equations are filtered first (although no details are given about the filter itself) and then the continuous filtered equations are discretized. But the two operations are not independent, the type of (implicit) filtering you have in the first step actually depends on the numerical details in the second step (i.e., resolution, order of accuracy, numerical dissipation etc.). Given this fact, a discussion of numerical issues in the context of LES modeling is quite relevant. A priori, this does not make sense since LES is presented as a model for continuous filtered fields.

The discussion here is driven by practical implementational aspects of LES. It has been shown that, with the same LES model, 2nd order scheme produces results that are quite different from those produced using a 4th order scheme. The difference is even larger if an upwind-biased 3rd order scheme is used.

Also, turbulent diffusion (as you have defined it) as represented by the turbulent or eddy viscosity does increase monotonically with "h" as long as the the wavenumber corresponding to "h" lies in the inertial range. And this is the case most of the time. If it is not, you are either conducting a DNS or VLES (perhaps even unsteady RANS). The scaling is nu ~ h **(4/3). In the limit of a DNS, nu reduces to molecular viscosity and h reduces to the Kol. length scale thereby retreiving the Kol. scaling law. This scaling is different from the h**2 scaling in the Smagorinsky model since |S| also scales with h (due to the fact that strain rates are computed using filtered data and a discrete approximation for the derivatives).

I agree with the points you've made completely. It is of course obvious that the numerical scheme has a crucial impact on the validity of the solution and perhaps even the model itself (especially since the model in this case is a function of the local length scale). Of course, as we change the resolution we change the accuracy of the |S| evaluations and thus nu_T, but how important is this (in terms of order of magnitude)?

The thread gave _me_ the impression that order of magnitude for errors in conjunction with the h^2 of the LES were being used (interchangeably) to justify whether a particular discretization order is better than another. I disagree with this, as a general rule.

I'm not familiar, but am I supposed to understand from your statement that accurate solutions are obtained in the following order: 2nd, 4th and 3rd upwind? If that is indeed the case, then you have already proved my point. If the order is 2nd-, 3rd and 4th, you haven't disproved my point, yet ) If the first case is true, then my point that it is the discretization scheme and not (necessarily) its order that makes a big difference is justified.

Also, your point with respect to LES turbulent "diffusion" (and I used it on purpose) being proportional to h is correct if we use a turbulent "diffusion" model - not at all obvious that this is the best model. What if backscatter was allowed in the model? (which is a clear problem with all Smag. based models, so far) But, perhaps I am/was digressing from the original discussion.

Adrin Gharakhani

It isn't a matter of what problem I was trying to solve, it is a matter of the ineherent numerical instability of explicit central differences for the convection terms. See Hirt J. Comput. Phys. 2, p. 339 for analysis of this method. My numerical experience supports Hirt's analysis: the code blows up when you run with small or zero viscosity. You can stabilize the calculation by adding enough viscosity (such as an eddy viscosity), but that just covers up a fundamental flaw in the method. The method can be stabilized by making it implicit, but that is a lot of extra work and the method still has problems with dispersive truncation errors. Even after looking at a couple of the Stanford papers, it isn't clear to me why they got good results from this method. I suspect there was something they did that was not reported in the papers (unfortunately that happens a lot in this business), and we will never be sure without asking the authors. I guess that's the long way of saying that I don't really know the answer to your question.