Interesting post!.

In my opinion, URANS is really powerful for truly transient problems, i.e: when we have external time-varying driving forces.

The frequency of turbulent fluctuations is a critical parameter. In other words, there is a threshold frequency below which RANS models could be used for predicting unsteady, in average, of turbulent flows.

On the other hand, we do not care much about the order of accuracy of such RANS - URANS methods. Whereas in LES that is fundamental. Thus, we should compare high order LES and URANS (time and space). To my knowledge, I have not read anything similar to this ( might be wrong though).

Finally, my opinion from my young experience as a researcher is that turbulence models are full of uncertainties that at the end of the day they add dissipation to retain the computation stable. (Take this last statement with a grain of salt)

What is the purpose of URANS in case of a statistically steady flow? I suppose that one should do RANS in such cases instead of URANS.

For understanding the difference in outcome, a simple example is the planar pressure-driven periodic channel flow. I did both LES and URANS in OpenFOAM for this test case and found that URANS resulted in a non-uniform instantaneous velocity field which was like a damped LES solution. In other words, I had to do time-averaging for URANS just like LES to obtain a statistical steady solution.

Another point is that we often use upwind schemes for URANS accompanied by a first-order temporal scheme. These together greatly affect the instantaneous solution and makes the comparison between URANS and LES less...meaningful.

As my final remark, in a practical perspective I do URANS when I need to have an approximation of the instantaneous flow field in a much shorter period of time.

Ehsan a time averaging in URANS makes no sense (at least based on the fundamental of RANS). Why do you need a time averaging for an Time Averaged solution procedure? I wonder why you needed the averaging.

Nevertheless, your input is highly appreciated.

Let me do a couple of practical examples.

1) the same statistically steady 2D flow problem (for example the flow over a cylinder) where the same statistically averaged BC.s are prescribed and both RANS and URANS are performed on the same grid. Why URANS should give a time-dependent field if no external time-varying force is present? Any type of statistical averaging should converge to the same averaging, and URANS should produce vaninshing time derivatives to converge towards RANS solution.

2) Consider the URANS and LES equations for a 3D problem, solved on the same grid, using the same second order space-time discretization. Forget the symbolism for filtering or averaging, just observe you solve exactly the same discrete equations each one having a forcing term. A forcing term for LES should take into account for filtering and SGS effect and a different forcing term for URANS should take into account statistical averaging and modelling. Now, apart from the filtering implied in URANS by discretization and grid (same as in LES) how do you take into account the statistics?

Interesting examples professor.

For 1) then I will lean toward the fact that URANS converges to RANS in a statistical framework. In that case, I would say there is no distinction between RANS and URANS. Therefore, I would be ignoring the difference between the URANS and RANS. In other words, why perform a transient computation if I I know that at the end they both converge to the same statistical state.

2) Interesting scenario. In reality, if URANS is performed on an LES mesh-like resolution, therefore the URANS should be similar statistically to the LES solution, but then this means there is no a clear distinction between the LES formulation and URANS. I do not know, I guess this is out of my experience/understanding. (similar conclusion from (1))

" the URANS should be similar statistically to the LES solution "

The question is: it appears similar (and someone must explain why) and we accept that or it must be theoretically similar?

Remember the example, same 3d flow problem, same discretization accuracy, same grid size. We have nothing else that the same discret form of the Navies-Stokes equations with a forcing term, that is the turbulence model. I can say for sure that if we call LES the implicit filtering provided by the discretization of the equations and the computational domain, this appear equally both in LES and URANS formulation. Therefore, I wonder if the forcing term in the URANS must give the global statistical effect on the solution, why we get unsteady and not steady RANS-like solution?

Conversely, if we accept that the unsteady solution in URANS is physically relevant and the turbulence model theoretically account for time fluctuations (I would see a rigorous demonstrantion) why we don't accept that this is nothing else that a form of LES formulation ?

Yes, thanks for the further contribution. What would be interesting is to invite any scientist to a "reverse engineering" process. Start from the discrete NS equations summed to the "turbulence model" as a unique distintive term and determine for each form of such term if you are solving either the spatially filtered PDE (LES) or the time-averaged PDE (URANS).

This is somehow an exercise resembling the determination of the modified equation.

I would be curious to find a theoretical justification that assess the role of URANS in terms of solution of a time-filtered PDE (that would be again that no statistical averaging is present and URANS cannot tend to the steady RANS).

P.S.: this issue is in my mind since many years ago but I raised now the question because of this impressive and computational expensive project that solved the flow problem in URANS formulation and was today an object of a webinar at Ansys.

https://www.cray.com/peloton-project-races-world-record

I forget to highlight this relevant sentence

" In contrast, LES arises out of space-averaging. So you cannot compare time averaged solution with space averaged solution and expect the same outcome!! LES or ILES provides time-dependent solution. "

Actually neither the space filtering nor the time averaging are really applied!! We just deduce them from the discrete form of the equations. And if the discrete equation differs only for the turbulence model, could we compare them?

" Don't we filter the full equation in LES for all terms? "

This is the key! The answer is yes but often is only a formalism in the PDE...

I mean that only if you adopt the so called "explicit filtering" technique you really have the action of a local filter. But the large part of the present LES code (all the commercial and open-source) adopt the "implicit filtering", that is the filtering is only a consequence of the distretization of each term and the computational grid but not of a filtering operation. And that is exactly what is used for the URANS formulation. Therefore, nothing but the turbulence model distinguishes an LES from URANS!

I think that it kind of is an idiosyncracy of all the turbulence modeling approaches, even explicitly filtered LES (after all, we are not filtering the actual NSE, are we?). It is LES that probably shed more light on this, making it apparent that RANS and URANS are flawed as well (having, basically, the same computational form). Actually is also kind of obvious (yet implicit) on their derivation/construction: we put tugether pieces obtained from time averaging analysis and fit them to an arbitrary eddy viscosity. The main fault for (U)RANS, I guess, is that there is no guarantee that the rest of the momentum equations will automagically satisfy the underlying hypotheses just because one (fictitious) term does. Yet, I don't think we actually have any practical choice.

" The main fault for (U)RANS, I guess, is that there is no guarantee that the rest of the momentum equations will automagically satisfy the underlying hypotheses just because one (fictitious) term does "

Paolo, assume that the underlying hypotheses are satisfied in order to be the URANS equations congruent to the statistical approach. Now, what in the world allows for a time-dependent solution in a statistically steady flow problem? Why the transient do not produce vanishing time derivatives if the formulation is congruent to the statistical averaging?

I can assume this time-dependent behaviour only if an external force produces a time-varying ensemble averaging. And when is no longer true that all statistical averaging are equivalent.

What really distinguish in the turbulence model that the numerical solution of URANS is so distinct from a LES formulation??

Have you seen the URANS simulation of the peloton ?

" Your point is that URANS is ILES with a turbulence model? "

Yes, and I wonder what else it could be when you look at the URANS solutions that appear so reach of details of vortical structures that we should not see in a real time-averaged approach.

So, in conclusion what is the real meaning of the URANS field we observe??

Dear Tapan, thanks for your attention. This investigation would give insight in this topic. In my opinion, the discretized URANS equations induce an implicit spatial filtering exactly as happens for the LES equations (as I showed in my JCP paper). The issue is to analyze the role of the URANS turbulence model: does it really induce a statistical averaging meaning to the resolved variable such that the spatial filtering has no longer relevance? And if that really happens, why the URANS solution does not converge towards the RANS solution in case of statistically steady flow problem?

Yes, of course, if one gets from running the transient URANS a steady RANS (a fact that I would theoretically expect) you have as a result only zero-th order statistics. Conversely, if the URANS solution remains unsteady we can compare high order statistics. The contradiction is that one must then really perform a statistically averaging on such solution, a fact that in my opinion is conflicting with the goal of computing directly a statistical solution.

Don't you think that RANS and URANS are worlds with some theoretical flaw in the practical numerical solution?

Filippo, I would distinguish between the issue discussed here and the peloton simulation, which might have had practical issues prohibiting an LES simulation. Of course, what I meant is that I agree with you: URANS should only produce unsteadiness compatible with the time scales allowed by the turbulence model. Yet, I'm not sure that, even in principle, such scales should be deterministic only, especially when several of them interact.

I think that the whole point is: 1) do we admit that URANS could in principle be applied for a turbulent flow around a cylinder or not? 2) if yes, do we really have a suitable URANS turbulence model for the task? 3) if yes, what we expect the solution to be? Steady? Unsteady? Why?

It is probably point 3 the real issue, in the sense that there is no agreed upon formal answer. Yet, the ensemble averaging formalism is clear on this: the solution must be steady in this and similar cases (i.e., the peloton experiment)

Paolo, the peloton simulation has a grid resolution comparable to the LES requirements. If you see the movies the small structures are described and they seems to survive for a time long enoug to be not compatible to a statistical approach ....

I agree but I gave a look at the article and the grid. According to my experience with Fluent they might have experimented instability with the central scheme needed for LES. Being forced to upwind they just picked up the only remaining meaningful choice (according to the main consensus in the field): URANS. I know this by experience with the code. I also know the group by literature (they also cited my work on LES inlet methods), they know what they're doing. Yet, the results perfectly highlight what we said. But this is a URANS problem at large, not for this specific case.

Of course, given this URANS ambiguity, the problem is that it is up to the single researcher to decide if unsteadyness requires LES or not.

Probably URANS also gained acceptance over the deficiencies of the LES wall requirements and before hybrid models were effectively developed.

Yes I see, I would have had more details if they tried to perform LES on the same grid and the problems they encountered. I also wonder if the numerical instability is due to the central unbounded discretization or something else.

In any case, the key is in understanding the type of variable one gets from such kind of simulation. I cannot see any theoretical reason why an ensemble averaging would not converge to the time-averaging in such statistically steady case. And since the discretized NS equations are the same, the only reason I see is in the turbulence model that acts as an SGS type rather than a statistical model.

Looking at the few grid pictures, it might have been the grid and the unbounded scheme together... Yet, they have my respect for taking home the grid and the simulation.

Concerning the main issue, in most cases the LES-URANS difference really boils down to the magnitude of the turbulent viscosity. Thus, basically, it is a matter of getting a sufficient amount of it in dependence from the bcs

In order to give a more rigorous framework for the discussion, I added in the main topic a two-page pdf with the equations.

So assuming equal equal resolutions and time-step (so discretization-related filtering is the same):

• No filtering is applied in LES. It is just like DNS (on a courser grid) complemented with a sub-grid-scale turbulence model to try to complete the energy spectrum with the scales that are not being resolved.
• The filtering in RaNS assumes the rate of change of the mean field is always much smaller than that of the turbulent field, and optimizes the "time-step" at each cell to fit that assumption. This way at all cells you get a statistically steady solution, but it may not be temporally accurate.
• When you're doing uRaNS you're forcing a common time-step to all cells: depending on the rate of change of the mean field, at each cell you're either getting: a) a statistically steady solution (if the period of all fluctuations is much smaller than your time-step) b) a time filtered unsteady solution (if the period of the fluctuations is either much larger or much smaller than your time-step) or c) something in between (at time-scales close to the time-step fluctuations are being smoothed, the mean field gets contaminated by the turbulent field).

So with uRaNS, to get a temporally correct unsteady solution you may have to go to a very fine time-step (which depends on the type of flow, naturally) to minimize your temporal averaging errors. At that point, might as well go to LES and correctly solve all scales (up to the limit of your discretization), with the added benefit of limiting the impact of the turbulence model.

- No filtering is really explicitly applied on the LES equations but the grid and discretizations do implicitly apply a filtering. Disregarding the explicit filtering formulaton, of course.

- RANS is a steady formulation by definition. There is no filtering, apart that induced by discretization that is largely disregarded by the effect of the model. There is also no time step to discuss about.

- In URANS you solve exactly the LES equations except from the term that take into account the closure model. Your intepretations a), b) or c) must depend only upon the turbulence model. If a "time.filtering" would exist it must be deduced only from the model. The only conclusion we can assess from the discrete equations is that a spatial filtering implied by the scheme is the same both for LES and URANS equations. You can work also using a very small time-step, there is no theoretical reason for which the URANS should not tend to the RANS solution. Remember that I am talking about flow problems that are statistically steady.

Sure, where I say "time-step" it should read averaging period when it comes to RaNS. However the point still stands: a turbulent flow is only statistically steady in relation to a relevant time-scale, what if that time-scale is not spatially homogeneous?

With uRaNS there will always be some inherent level of temporal averaging, associated to the closure model just as you say. So yes, in a statistically steady flow problem uRaNS will revert to the RaNS solution.

However it will differ from the LES solution for that same reasoning, no?

Regarding the spatial filtering I agree: no difference between LES and uRaNS.

> what if that time-scale is not spatially homogeneous?

Do you mean when there is no unique characteristic lenght scales?

> However it will differ from the LES solution for that same reasoning, no?

Yes. It "must" differ by definition of the continuous PDE where in the LES one you have a local spatially filtered variable and in the URANS one the statistically averaged. The issue is if you look at the discrete equations (see the PDF I attached in the main): does the URANS closure model by alone returns the meaning of statistical averaging into the resolved variable? And if that happens, why URANS does not converge towards RANS when there are no external time-varying forcing?

Or the discrepancy depends on the wrong setting of the BC.s?

I see what you wrote about such errors, as you wrote owing to this error different Fourier components will travel at wrong wave speed producing changes in the packe waves, that is dispersion. Generally we can find in literature several analyses for simple linear advection equation and some non linear model such as the Burgers equation. The differences in real problems between "wiggles" and physical flow structures are considered much less in the papers. Maybe because it is assumed that if you perform LES the dispersion error has less effect owing to the dissipative effect of the SGS. In DNS, if the grid resolution is correct, the physical dissipation should cover this dispersion, too.

I think that the global modified PDE (in physical or wavenumber space) would highlight the role of the dispersion. For NS equations this is not so simple, for example years ago I studied the local truncation error due to time integration of the NSE based on the fractional step.

The question is if URANS solution is unsteady as a consequence of a mix between numerical errors and stabilizing model effect. Again, I would be happy to see the solution for the flow over a cylinder where exactly the same time-space discretization is used and only the turbulence model will distinguish between LES and URANS solutions.

I hope to get collaboration on this topic.

I can say that one of the sources of dispersion errors is in the discretization of the non linear term. The advection quasi-linear form is one that produces dispersion.

Thank you all above for this very interesting discussion. I hope you allow me to join. I have been giving this topic some thought as well over the past years, in the framework of the shallow water equations and the formation of dynamic eddies in URANS computations (or to be precise U-SWE computations) with steady boundary conditions. In other words, any dynamics that emerged in these computations came purely from the nonlinear excitation of discretization errors (which is I guess the topic of this discussion).

In these investigations, I experimented with the effect of the nonlinear advection scheme and the grid resolution on the emergence of eddies. Results indicate that most computations are performed with a 'numerical' Reynolds number which is not at all linked to the fluid viscosity (at least for practical grid resolutions), but to the artificial viscosity, effectively emerging from truncation errors from the discretization (and mostly from the nonlinear advection discretization). For numerical Reynolds numbers that are low enough (coarse grid, 'poor-quality' advection scheme), the results were steady and only resembed the time-averaged solution. For higher grid resolution and/or better advection schemes (not necessarily higher order, but with better conservation properties), the solution became time -dependent with vortex shedding (flow around a cylinder, flow around groynes in rivers).

Due to its link to the numerical viscosity, the aforementioned numerical Reynolds number is automatically space-, direction- and (in case of flow dynamics) time-dependent, resulting in an artificial/numerical solution.

For commonly-applied grid resolutions in practical environmental flow applications, the numerical Reynolds number is orders of magnitude lower than the theoretical one, based on the fluid viscosity. However, - and this is the main thing and also the reason why ILES methods were developed - this artificial solution resembles in some/many cases what happens in reality. In other words, despite the numerical/artificial nature of the (numerical) eddy viscosity, the flow dynamics and eddy formation, shedding frequency, etc. shows good agreement with reality, at least for certain advection schemes that exhibit certain conservation properties (momentum or energy).

To come back to the topic question:

In my opinion, URANS with/without a turbulence model and LES without explicit filtering differ really only in the turbulence model. The dynamic effects seen in URANS are artificial and in both models greatly depend on the grid resolution and the discretization of the nonlinear terms, whose discretization errors cause dispersion and spurious modes, which might look 'physically-correct'. The question for me then remains: do we except this and say that the results are valid despite being obtained for the wrong reasons in URANS (discretization errors), or do we strive for a correct solution that was obtained for the right reason (like in LES)? And perhaps linked to this: how do we know for sure whether a certain method always provides adequate results, for the wrong reasons?

My apologies for this long response.

Best wishes,

Frank Platzek

And sorry for posting twice ...

Duplicate posts can be erased by clicking on the right where the "time from the post" appears and a small arrow down is

My opinion is that the issue is somehow more complicated than appears... I don't think that the key is only on the equivalent Reynolds number that is in effect from spatial part of the local truncation error. We could use centred spatial discretizations. And that is both for LES and URANS equations.

I address you a fact that I observed and have no clear explanation for that; usually in doing LES we perform a simulation to be used as comparison without any SGS model (LES no-model or under-resolved DNS). I have never seen someone trying to do a URANS without a turbulence model. Why? Obviously, that would be by construction of the discrete equation, nothing else that a LES no-model solution! So, the URANS equation is a sort of ILES without model and becomes a statistically averaged solution with a model? That sounds strange in line of principle...

Dear Frank

this statemet is relevant:

" In other words, despite the numerical/artificial nature of the (numerical) eddy viscosity, the flow dynamics and eddy formation, shedding frequency, etc. shows good agreement with reality, at least for certain advection schemes that exhibit certain conservation properties (momentum or energy). "

The conservative property in the discrete form is fundamental as it improves also the correct waves propagation. That does not happen for the advection form of the non-linear term. It can be shown that also the local truncation error retains the conservative form if the divergence (or integral) formulation is used.

A comparison is shown in the book of Leveque about FV for hyperbolic system where the solutions for the Burgers equation shows the behavior for conservative and non conservative formulation.

However, for the aim of the main discussion, we can always assume the same conservative form for LES and URANS codes. This way the numerical effects are the same.

Thanks, Frank Platzek fro such excellent explanation and bringing in another perspective to the topic. I am eager to read more about this and a formal evaluation is definitely mandatory from the experts in the field!

Professor Denaro's final comment hit right on the nail and it drove me to the conclusion that there is no such distinction between LES and URANS at least under the assumptions of this post. Somehow is very disappointing to see that decades of work don't hold water since the true difference is a turbulent model and not the framework of the formulation.

I was also thinking about the dispersion-dissipative errors, but we assume that both formulations suffer the same type of errors. Owing to the fact that both formulations are discretized using the same methodology.

My humble opinion (I can be completely wrong) is that the future is in the unresolved DNS solution and not the type of formulation we use nowadays. Also, Professor Tapan mentioned something important about the so-called "turbulence-like" structures. I do not know if the future is in those models, but I really doubt it. A couple of weeks ago I told my advisor "Professor, I think we will need to go back to the drawing table in the near future if we want to see a breakthrough in CFD". I am glad I said that after graduating but that is my personal opinion. There is to much noise in the area that it is sometimes difficult to keep us focused. But again, my opinion is as a young researcher and not as an expert!

I think that using the simple 1D Burgers model one can start to analyze the first issues...

If we look at the matter from the other side, what makes URANS necessary, with respect to RANS, in the case of steady bc?

In the current industrial practice, it is just lack of convergence.

One may then argue, instead, that it is just an improper RANS model formulation for the given case, which would not be surprising, considering the origin and nature of such models.

After all, for any statistically steady flow we can, inprinciple, compute the mean Reynolds stress tensor and supply it to a RANS computation that, if suffciently accurate, should converge to the underlying steady flow (shouldn't it?).

If such super RANS model would be available, I suspect that the unsteady term would eventually vanish when it should or easily track bc unsteadyness (as the remaining part would be well balanced by the super model). But I am afraid this is not so easy to prove in practice.

I agree, non convergent RANS produce a residual that is simply converted into “physical” time derivative. That is sufficient in the industry to justify the results...

Professor Denaro, Paolo,

if that is the case why don’t we just focus on URANS and stop using RANS? At the end of the day a temporal approach is less stiff from a mathematical stand point. The problem that Paolo brought up is widely seen but still, industry is still using such approach. I guess that using a temporal approach is not expensive anymore with such powerful computers (if and only if numerical implementations are enough efficient and leverage computational architecture)

Dear Julio,

if the problem has a physically statistical state and RANS does not converge to a steady solution, the problem is in turbulence model or in the numerical solver. Now trying to interpret such non-vanishing residuals as physical time derivative is quite common and drives to denote URANS such solution but what is the theoretical justification for that? Again we see that only the turbulence model is responsible of the meaning of the resolved variable (apart from the spatial filtering effect of the discretization). Therefore, I would expect that the turbulence model between RANS and URANS are totally different, this latter should taking into account for a virtual time-filtering (not a statistical averaging) applied on the NS equations. But I am not able to see how the URANS model highlights such filtering.

Using again the example of the recent paper studiyng the peloton of cyclists, they need to explicitly compute a statistical time-averaging from the URANS solution! That would mean that the non-vanishing residuals in RANS have a vanishing time averaging.

Is'nt that quite a confusing framework?

The theoretical assessment can be accepted if the URANS model shows the role of the time filtering on the equations. This way we can say that URANS is not based on a statistical approach and does not need to converge to the RANS. Is a sort of reverse enginering..

Is someone able to demonstrate that rigorously?

Maybe Paolo has more comments

Filippo, I don't know if I got correctly your comment but, actually, I would say the contrary. If the super-duper ideal RANS model I mentioned in the previous post really existed, I would expect it to be INDEPENDENT from anything related to the unsteadyness... thus, in a certain sense, just a regular RANS model as we know it today (but actually working in all cases).

I think this is fundamental, in order to then correctly interpret the unsteady version, URANS. In this case the temporal term would really just follow only deterministic unsteadyness, explicitly introduced by the user via bc or source terms (or similar)...

...or, at least, I would expect it to do so, because I know it to always converge in the steady case (so that model unfittedness is not randomly projected into the unsteady term, working as a local acceleration in the end).

As a side note, LES also experiences such random projection of model unfittedness into a local acceleration (which makes the difference between a priori and a posteriori testing); yet, we blindly accept it, probably because the unsteady term is the natural bin for anything wrong

Not sure about your idea of "super ideal model"... My opinion is maybe ideal but I would immagine a test where I can get first a DNS solution and compute all the statistical terms to close exactly the RANS equations without introducing the model anche check if a steady state is reached. Then I would try the differences using the practical turbulence RANS model.

Unfortunately, the same procedure is not rigorously correct to check the URANS: we don't know the actual type of averaging in the discrete equation and the model computed from the DNS does not correlate in time with the URANS variable.

On the other hand if you consider unsteady BCs. that would make somehow more rigorous the meaning of the URANS. But you should provide the unsteady BC.s based on the proper time-filtering, that is such BC.s are not the same used in DNS or LES.

Finally, coming back to the equations I wrote in the pdf file, how do you consider that the discrete equations are exactly the same? How can the time derivative be a bin for space filtered, time filtered or residuals?

I expect that the turbulence model must explicitly contains such operators to distinguish the formulation. At the best I can understand LES does that.

Can we try to use also a simple URANS closure model to see this fact in the simple Burgers equation?

Yes, that's the idea of my super ideal RANS model (Giuliano De Stefano has done something similar in LES, if I recall correctly, calling it perfect SGS model, or something along these lines). Imagine doing this with the cylinder flow. Then I expect that both RANS and URANS with such a model would give the same results, URANS specifically becoming steady.

In theory, I don't see it as particularly troublesome doing this on such a steady case nor on an unsteady one (because of the bcs). Even in the unsteady case, the idea is that what you put in as unsteady it comes out in the unsteady term. You can always check this afterwards. If some spurious frequencies are present, something is wrong. Yet, in practice, there might not be any code to actually do this for a significant case (maybe Nek5000).

Concerning the original question, and summarizing my previous toughts, I see URANS feasible only when the time variation is suffciently slow to have each time step independent from the other ones. You would need the super ideal RANS model to do this, in order to avoid any energy introduced at the largest scales to spill down toward the smaller ones. In this scenario, I think there is nothing troublesome in itself that URANS and LES have the same equation. Of course, this does not mean that I think LES, RANS or URANS actually fullfill the promise of producing results congruent with their modeling assumptions. For the moment, I just accept that 1) this is how simulations are possible/intended and 2) most models simply fail at this (you know there is a lot of literature on LES concerning the filter induced by the SGS model, yet its relevance in this context has probably not been explored).

I don't think 1D Burgers, or anything non NSE, could be relevant here. You would introduce more difficulties (there would be a lot of complex things to address for the first time in such framework) than those you try to remove with the simplification.

Ok I agree about the perfect model for RANS computed from DNS that should always provide a convergent solution. But in URANS the way in which you compute this “perfect” model depends on the type of averaging underlined by the URANS equation. And what about that? If you assume a time-filtering you admits the possibility of an unsteady solution also for steady bcs.

Many thanks Tapan, I downloaded the paper and will have a look.

In my opinion if you extract from DNS data the statistally averaged fluctuation tensor, as you get a tensor that is no longer time-dependent, you can use it as "perfect" model both in RANS and in URANS equations and you should get the same convergent solution. Don't you think?

Conversely, if you assume that the URANS equations are based on a time filtering, as the time-filtered fluctuation tensor (computed from DNS) is still a function of time, using it into the URANS equation would correctly drive to an unsteady solution.

In principle a perfect model contains the answer. But what when we use a classical model approximation in URANS, does it really contains the information of a time filtering?

I agree about the ambiguity of turbulence models. However, we can start someway from an hystorical perspective. For example, in terms of URANS the idea of a mixing lenght should'nt be related congruently to the introduction of the time step and a time-varying characteristic velocity to take into account (possibly) the role of the unsteadiness?

However, I read your paper, from your DNS you could give a first answer to the issue: computing the statistical averaging of the fluctuations and using this "perfect" closure to the steady RANS. Does the solution converge or you get oscillating residuals mimicking the time derivatives of URANS?

I would add a further question to the discussion. Do you know about some published papers that analysed, for the same flow problem, URANS and LES using different BCs (time versus spatially filtered)? For example, an inflow condition on a plane. Formally, that would be a key in the difference between the two solutions

I have always wondered the effect of the BC on the URANS. I have also asked the same question to Professor Denaro before, What happens if we use a LES "like" Inflow condition in a URANS formulation. However, how do we impose the turbulence model quantities such as K-E or any other method? Unfortunately, I have no found anything like this before nor a published paper.

Julio, recently some papers at CTR analysed the BCs for LES in terms of filtered condition rather than natural condition as in DNS.

Therefore, once the concept of the local time-averaging implied in URANS is assessed, one could proceed similarily

In my opinion, another aspect or point of view is important. Namely, the fact that the original unfiltered and unaveraged Navier-Stokes equations contain only a diffusion term dependent on the kinematic viscosity. There is no 'turbulence-term' present. The turbulence emerges from the nonlinear interaction of different scales in the convection term.

If we then look at the RANS equation without a turbulence model (i.e. only kinetmatic visosity), these are exactly identical to the original equations, i.e. the Reynolds-averaging is only present in the definition of the u-velocity, being the actual velocity or some averaged velocity. However, in the theoretical limit, where the spatial resolution and time step are reduced towards zero, the answer of this RANS model should converge to a DNS result, at least in my humble opinion.

To me, this means that the Reynolds-averaging is only a concept that one has to keep in mind when looking at under-resolved results (in space and time). The same would hold for an Implicit LES approach, under the condition that the discretization errors / numerical diffusion terms tend to zero fast enough with increased spatial and temporal resolution.

For LES with an explicit filtering approach I have too little experience and I cannot make such a comparison. Perhaps someone has an idea?

Just to add to the discussion on boundary conditions:

Similar considerations (concerning the 'resolvedness' of the results) then hold for the boundary conditions. My belief is that the boundary conditions strongly depend on the chosen grid resolution and chosen time step. When chosen improperly (too coarse or too fine) they might erroneously steer the results in the wrong direction (at least if the internal solution is influenced by the unresolved turbulence present in the boundary conditions). Any resolved turbulence should develop in the interior of the domain, when the boundaries are sufficiently far away from the area of interest.

Tapan, as you know the school of Stanford is based on the classical filtered approach, they do not encourage the ILES idea...

The idea of setting filtered velocity as BCs leads for example to a slip condition on the wall, see Article Investigation of the slip boundary condition in wall-modeled LES

According to such idea one could generalize the concept of a BC for the velocity that depends on the type of formulation

Frank, in the RANS the difference is clear and is implied by the fact that the steady equations are solved. Without a turbulence model, the RANS can be seen as the steady NS equations, therefore valid only for specific laminar steady flow. Conversely, in URANS the role of the time derivative changes this framework. Unfortunately, for time and space steps going to zero the effect of the turbulence model can be non-vanishing and you do not get convergence towards DNS as we accept in LES formulation.

Explicit filtering in LES is analysed in some published papers but is not a common procedure as it requires a quite fine grid. I am working on a formulation that is somehow linked to the explicit filtering as in the integral formulation you compute a surface integral of the resolved convective term.

" Similar considerations (concerning the 'resolvedness' of the results) then hold for the boundary conditions. My belief is that the boundary conditions strongly depend on the chosen grid resolution and chosen time step. "

When we adopt an implicit filtering in LES, we can for sure say that for vanishing spatial step sizes in the three directions the filtered velocity converges towards the unfiltered one (the DNS one). Of course that is no longer true if only one step tends to vanishes. Therefore, the condition at a wall for a filtered velocity depends on the width of the filter.

In URANS we could think in a similar way in the spatial filtering effect. But if we assume that the unsteady behaviour is formally accepted by the time-filtering assumption, what about the supplementary time-filtered BCs? It seems that URANS cannot adopt the same BCs used in RANS but it seems that also some difference with LES appears.

Sorry, where I wrote RANS, I meant URANS of course. Thank you for noticing!

I attach a page to further add a rigorous framework for the questions

Let me add more comments on this interesting issue.

In general, we have the approach chain: steady RANS - URANS (VLES) - hybrid RANS/LES techniques - pure (explicit) LES - implicit LES - DNS.

Indeed, all of them in the initial differential form, after filtering/averaging procedure and closure assumption introduction, use the same NSE plus the extra forcing term of turbulent stresses defined by a particular turbulence model. This form "forgets" the initial step of model derivation with filtering/averaging and "remembers" only the specific functions for turbulent stresses. Additionally, the discretized NSE have the numerical dissipation/diffusion effects which depend on grid size.

As a result, the discretized NSE have the effective viscosity = molecular + numerical + "turbulent" + "apparent" viscosities, where:

- molecular viscosity stays everywhere, being 1/Re in non-dimensional NSE;

- numerical viscosity is quite small in DNS in comparison with molecular one, but is essential in ILES;

- "turbulent" viscosity is absent in DNS and ILES, but is the leading term in all other tools, including URANS and explicit LES;

- "apparent" viscosity appears in models with turbulent stresses defined by separate differential equations rather than by algebraic relations, and simply replaces the "turbulent" viscosity, working by the same manner.

The effective viscosity produces the effective Reynolds number in NSE. The numerical viscosity is reduced with grid size, as well as the "turbulent" viscosity in LES since it contains explicitly the grid size. However, "turbulent" viscosity in both RANS and URANS is not reduced and stays the same for all grid refinements. This is the key difference between LES and URANS. Moreover, this feature implies that both RANS and URANS can have the essentially grid-independent solutions at fine enough mesh. On the other hand, both LES and DNS have no grid-independent solutions for instant fields (we need first to perform the time and/or space averaging, then check grid independence of averaged solutions).

Next, steady RANS solutions can simply be obtained by time relaxation of time-dependent NSE with a RANS model for turbulent stresses. I did this in channel flows with backstep and square rib. The absence of steady RANS solutions, e.g. in a wake past a circular cylinder means that the efficient viscosity (equal to turbulent viscosity at high Re for quite fine grid) is not enough to suppress low-frequency instability induced by the flow geometry. Perhaps, presence of the bottom wall in the backstep flow helps the flow to eliminate such an instability, whereas absence of walls for a free shear flow past a cylinder provokes the development of vortex street with low frequency of perturbations. The remaining physical instability associated with coherent structures in the mixing layer above the recirculation zone behind the backstep has higher frequencies so lay within the workable area of the actual turbulent viscosity. It is obvious that grid-independent efficient viscosity of URANS, nevertheless, is not capable to remove instabilities of all frequencies from the flow.

Finally, the origin of such an instability in both URANS and LES computations is the same. With steady BCs the code always has round-cut and discretization errors, and numerical instability can grow, even if it is not assigned in initial or boundary conditions, but if the flow geometry allows it, and the efficient viscosity is not large enough to suppress strong low-frequency perturbations in some areas.

I hope, as an ideal turbulence model eventually we will see the UNIFIED approach based on the hybrid RANS/LES techniques (some attempts are in progress in the literature) which has the blending function gradually switching the flow from (U)RANS mode to LES mode where it is appropriate. This is because in the foreseen future no one could demonstrate pure LES or DNS applicability for very high Re flows, e.g. around a plane or above a city.

Thanks Sergey for your contribution, it highlights again that the interpretation of the solution for the different formulations is not so straighforward.

As a further example, just consider an inverse procedure: a database of unsteady 3D veloctity fields for the solution of the flow over a cylinder. Without knowing anything else do we have a rigorous protocol to say that is a LES (implicit or explicit model) or URANS database? If URANS is based on the localized time averaging, the turbulence model should have a vanishing contribution for the time averaging width going to zero and a refined grid, that is the solution should tend towards the DNS one. Conversely, if URANS is based on the statistical ensemble averaging it must converge always towards RANS even for a DNS grid.

Does it really exist in the practice such distintion in the URANS modelling? It would be the link to go ahead in the hybrid modelling that automatically switch.

First, I had only experience in steady RANS solutions obtained by time relaxation of unsteady NSE + turbulence models, as well as that for explicit and implicit LES (DNS) where only unsteady solutions were obtained, of course. No experience yet for URANS or hybrid RANS-LES techniques, only reading of papers. However, this is an evident way to move forward in future studies.

Second, from a database of unsteady 3D veloctity fields for the solution of the flow over a cylinder we can simply estimate the scales of typical structures viewed e.g. from Q-criterion distribution. Generally, DNS, LES, DES and hybrid RANS-LES tools should give very fine structures, whereas the URANS method simply based on a classical RANS model will allow only very-large-scale structures with very low typical frequencies which survived after suppressive and smoothed effects of turbulent viscosity.

Third, there are the second-generation URANS models which are quite far from classical RANS assumptions (see e.g. Article Hybrid LES/RANS methods for the simulation of turbulent flows

Fourth, as said by you and others, final URANS model formulation 'forgets' the averaging type and remembers only the final algebraic functions or differential equations for turbulent stresses. It is obvious, that DNS or ILES solution can not be obtained within the URANS approach since the latter always contain non-zero contribution in turbulent stresses for any grid and time steps whereas the former has zero turbulent stresses.

First, my assumption is that LES and URANS are performed with the same discrete method and the same computational grid and time step. This way, the impact of the local truncation error is the same as well as the effect of an implicit filtering.

As a consequence, the meaning of the resolved field is just due to what the model implies. If we assume that the model in URANS does not vanish in a DNS-like resolution, this is a correct response of the turbulence model to an ensemble averaging, as same as happens in RANS. But that should also drives the time derivative to vanish and the solution tending towards a RANS one. Conversely, if the time derivatives in URANS do not vanish we can conclude that either the model is not congruent to the ensemble averaging or there is a numerical issue in the convergence (again like in RANS). The other way is that the field is not a statistical ensemble averaging but a localized time averaging and the model is able to do such a response. But at this point I can say that we can introduce any model that is not eddy-viscosity dependent. For example, I can think of a scale similar model in time, resembling the Bardina model.

As you see, I continue to see a lot of contradictions in interpeting the results...

Tapan, you are correct but I assume that the impact of the numerical error, if present, acts in the same way in LES and URANS: same discrete method, same computational grid and time step. As a consequence all we can deduce to distinguish the formulations is in the forcing term. It either vanishes or not in a DNS resolution.

Sergey,

" from a database of unsteady 3D veloctity fields for the solution of the flow over a cylinder we can simply estimate the scales of typical structures viewed e.g. from Q-criterion distribution. Generally, DNS, LES, DES and hybrid RANS-LES tools should give very fine structures, whereas the URANS method simply based on a classical RANS model will allow only very-large-scale structures with very low typical frequencies which survived after suppressive and smoothed effects of turbulent viscosity. "

In principle, the Q-criterion can highlight the smallest structure (with some uncertainity) but also in LES you can have that the combined effect of smooth filter and eddy viscosity models can produce that same effect of suppressing scales of the motion at high resolved wavenumbers. This is typically seen in the energy spectra where you see an "artificial" dissipation range close to the cut-off frequency.

I think that the correct issue is in you statement "URANS method simply based on a classical RANS model" !

Why a URANS that really want to describe a physical time-depended behaviour of the flow should be based on a classical RANS model? If it is based on such formulation, this URANS has to produce only steady solutions! Of course, I am always discussing the case of stationary flows.

Filippo, let me add some comments to your statements:

"LES and URANS are performed with the same discrete method and the same computational grid and time step. This way, the impact of the local truncation error is the same as well as the effect of an implicit filtering."

Agree with this. We have discretized forms of all NSE terms (time-dependent one, convection, viscous diffusion, pressure gradient, body force terms, turbulent stresses, etc.). If numerical schemes are the same, the only difference comes from the discretized turbulent-stress term.

Next, I suggest you to ignore the (ensemble or time or space) averaging ways in RANS, URANS or LES, and look only at the final constitutive relations applied to close the original NS equations, i.e. to parameterize the unknown turbulent-stress terms created from the part of convection term in both (U)RANS and LES. For URANS such relations should still not contain grid units, so are entirely grid-independent. For LES the subgrid-scale tubulent stresses use SGS eddy-viscosity models like the Smagorinsky-type assumptions directly involving grid sizes dx, dy, dz. This means that SGS eddy viscosity will become smaller at finer grid, whereas the eddy viscosity in RANS and URANS will stay the same for any grid size, even in the DNS mesh limit.

Recall, the effect from numerics of other NSE terms becomes smaller at finer grid, so "numerical diffusion" with grid refinement can eventually be neglected for URANS (in comparison with "turbulent diffusion"), but not for LES.

Finally, there are hundreds of RANS closures, and their variety and complexity arises from the strong anisotropy of large-scale motions (i.e. that of normal Reynolds-stress-tensor components) in different turbulent flows. On the other hand, the smaller-scale motions (turbulent eddies from inertial and dissipative ranges of spectra) are assumed to be more isotropic, so SGS eddy-viscosity models are usually simpler and often appear in the mixing-length form only. However, the values of SGS eddy viscosity at fine enough grid in LES are much smaller (so the error in it will not so visible in explicitly resolved eddies in LES) than the eddy viscosity in URANS at the same grid. This is why we do not see unsteady small- and medium-scale motions in URANS, except those in flow pictures produced by the second-generation URANS like SAS and PANS where the eddy viscosity can artificially be reduced at some conditions.

Nevertheless, large-scale periodic motions can still arise in unsteady NSE with RANS models if these motions have typical time periods much longer than those for all turbulent fluctuation motions, i.e. for all turbulent eddies. The examples are not only tide and daily breeze motions in sea shoreline, which have unsteady forcing in BCs, but also internal waves having no unsteady BCs, but produced due to wave-like behavior of unsteady NSE with buyoancy terms. The example with periodic vortex motions in the wake behind a circular cylinder in URANS also represents a sort of long-period non-turbulent oscillation arising now from the rest of convection term, although if the eddy viscosity in the RANS model would somehow be much larger it could suppress such an oscillation too to get the completely steady solution.

Sergey,

I agree to your observations but there is a point that is the key of the whole discussion:

" Nevertheless, some large-scale periodic motions can still arise in unsteady NSE with RANS models if these motions have typical time periods much longer than those for all turbulent fluctuation motions, i.e. for all turbulent eddies. "

To justify this statement, the only way is to consider the resolved variable in URANS not as a statistically averaging as in RANS (in no way this kind of averaging, supplied by classical RANS modelling can provide any physical time variation in the variable while can produce a lack of numerical convergence) but as a time-averaged variable (for example as described in the book of Wilcox). This way the resolved variable is able to do what you wrote. But that implies that a non-classical RANS model must be supplied but a model containing the time-filter width must be introduced. As a consequence, the reduction of this time filter width implies that this kind of URANS would tend to the DNS, contrary to the classical idea in the URANS.

What I would say if that we consider the classical URANS for statistically steady flow we can only accept a steady solution congruent to RANS. If we want a physical meaning of the time fluctuation we must think differently to the resolved variable: it is more a LES-like in time and the model must contain such information.

Do you agree?

For example, this paper shows the URANS equations for a statistically steady flow over the cylinder (if you perform an averaging from DNS you get a steady averaged solution) where the time-dependend velocity U_bar is solved by using a phase averaging for the Reynolds tensor .

http://www.itam.nsc.ru/users/libr/eLib/confer/ICMAR/2014/pdf/Palkin%20et%20al_221.pdf

Again, such averaging are never really adopted, they are implied by the turbulence model.

This other paper has, in my opinion, a strictly adherence to the concept discussed about the meaning of the URANS variable.

And it is clear from the results why a RANS-like model as the eddy viscosity lenght makes the variable tending to a statistical one while using a more complex model, such the k-omega (where frequency variation are considered), the variable has more the meaning of a time-filtered one.

Just as a further hint, I was thinking about the LES vs. URANS equations if we cancel out the explicit turbulence model. In the former case, we simply go in the well known ILES methodology, so why in the latter case no one denotes that as Implicit URANS? And, actually, without a model the equations are perfectly identicals, therefore in which way could distinguish ILES from IURANS? Should we consider the local truncation error of the time derivative similar to the idea of ILES that considers the spatial error as implicit model?

Filippo Maria Denaro, let me add some fragments from sections 2.1 and 6 of the review (Frohlich and von Terzi, 2008) mentioned earlier (https://tu-dresden.de/ing/maschinenwesen/ism/psm/ressourcen/dateien/mitarbeiter/froehlich/publications/Froehlich_vonTerzi_Hybrid-LES-RANS_PAS_08.pdf?lang=en). These experts discussed various branchs of hybrid LES/RANS methods, so were "thinking about the LES vs. URANS" a lot and understand well the difference between these methods:

" 2.1. Unsteady RANS.

Models employing the RANS equations are based on a definition of a mean. ... The averaging operation is applied to the Navier–Stokes equations yielding equations governing the mean motion of the flow. These equations contain an unclosed term which is replaced by the RANS model. ... For statistically steady flows, the temporal mean is an appropriate choice. For flows with slow variation of statistical properties (slow compared to the characteristic turbulent time-scale) a finite-time temporal average can be used [24]. For unsteady flows with some basic frequency, a phase average can be introduced. In this latter case a triple decomposition ... for the velocity vector was proposed in [25] ... It has become common to name RANS modeling as URANS whenever the computed solution is time-dependent. The approach then is to apply an existing RANS model and to aim at resolving some of the unsteady features of the flow without recalibration of model coefficients.

With respect to large-scale unsteadiness (large in space and time, as opposed to turbulent fluctuations) it is useful to distinguish between two cases. ... A second case, comprises situations with internal instabilities

of the flow, such as bluff body flows. In the near field, scale separation usually does not hold: The very largest vortical structures depend on details of the transition process (influenced by the thickness of some boundary or shear layer, etc.) and disintegrate into smaller and smaller structures farther downstream. In such a situation, phase averages can be constructed and the terms in the RANS equations can be properly defined. However, a substantial amount of interaction between turbulent fluctuations unresolved by the URANS approach and the resolved fluctuations occurs which is delicate to handle. It is the second step, devising a model for this situation, which poses the problem. An unmodified RANS model is likely to be unsuitable for this task.

A good illustration of these arguments is provided by the flow around a square cylinder ... The URANS results considered here do not yield the correct Strouhal number St. This frequency, however, is a very insensitive quantity for most bluff body flows. Hence, if not even St is correctly captured, no confidence in the results can be attested at all. ... The example given here illustrates the difficulties traditional URANS calculations may encounter. On the other hand, URANS simulations can be substantially more successful in determining the mean flow than a steady RANS computation [28]. In our opinion, the main concern when applying URANS is the confidence one can have in the results when no experimental data are available for validation. Sometimes grid convergence is not achieved in the range of commonly employed grid resolutions [29]. The use of URANS can hence only be advocated in cases of clear scale separation as described in the beginning of this section. When this is not the case, as in the second situation discussed, the approach seems delicate.

...

6. Second generation URANS models

RANS models are models only involving physical length-scales. LES models, in contrast, have been classified above as models containing, explicitly or implicitly, a length scale related to the numerical grid. This length scale determines the size of resolved fluctuations. Very recently, models have emerged which aim at resolving a substantial part of the turbulent fluctuations but do not contain such an explicit dependency on the computational grid. Consequently, we term these models second generation URANS models (2G-URANS). The essential characteristics, additional to the independence from the grid scale, is that the model contains a term sensing the amount of resolved fluctuations (temporal or spatial). This is in contrast to the classical URANS procedure described in Section 2.1. "

From these fragments, one can conclude that URANS should be used with the great care, and LES is preferred. To improve the situation, some hybrid RANS-LES tools and advanced second-generation URANS techniques reviewed by the authors were proposed.

The key definition (stressed by bold) means that LES models do contain grid-related scales, and URANS models do not contain any quantities related to the numerical grid. This immediately answers your question "why in the latter case no one denotes that as Implicit URANS?" You can see that ILES can indeed exist, but IURANS is impossible: the latter would contain grid-dependent quantities via errors of numerical discretization schemes which contradicts with the (U)RANS definition. Another consequence for (U)RANS models: someone can eventually get the grid-independent solution, performing the series of computations with successive grid refinement, because turbulent stress models of (U)RANS do not depend on scales related to the grid. However, grid-independent solution can not be found for filtered LES fields, because SGS stress models depend on quantities related to the grid (perhaps, this is possible only after LES data averaging on a relatively large time period).

On the other hand, I have just discussed the problem "URANS vs. LES" during the local conference (http://conf.nsc.ru/icmar2018/en) with the experts Andrey Minakov, S. A. Isaev who have been using successfully the different URANS models to examine unsteady flow features, like Palkin et al (2015) mentioned earlier by you. I may give later some further considerations and deeper insight what really happens in URANS computations of a flow around the circlar cilynder.

Filippo, let me give here a brief review of the mentioned paper on URANS computations of a flow around the circlar cilynder (THMT15_E331_Palkin_et_al.pdf), with some outcome for the issue "URANS vs. LES".

In the paper, for such a flow at Re ~ 103--105 both two different models (a two-parameter EVM k-w model and the full differential Reynolds-stress model) give a picture of unsteady vortex structures which is quite similar to LES. The identical spectra for URANS and LES calculations are especially striking (Fig. 7b). It would seem that this can not be expected from RANS models, because the effective viscosity produced by these models should successfully suppress all fluctuations, in contrast to a relatively small subgrid-scale viscosity of LES, which suppresses only fluctuations of small scales. In contrast, we see, at least, two modes of low-frequency unsteady motions, including the vortex street shedding and the Kelvin-Helmholtz vortices in two mixing layers above and below the cylinder, as well as some complicated 3D structures (Fig. 8).

To understand the essence of what is happening in the (U)RANS runs past the cylinder, the following sentence (p. 3 of the paper) should be taken into account: "inflow was determined by the uniform velocity U∞ and zero free stream turbulence". This means that a homogeneous laminar flow is set upstream of the cylinder. That is, in fact, the ILES equations (= DNS with an unresolved grid) are artificially applied around the cylinder in both RANS and LES calculations. In this case, one actually can expect the generation of instability during the laminar separation from the top and bottom of the cylinder, which is triggered by rounding errors and other numerical perturbations. A turbulence model is switched on only when the sufficiently large turbulent kinetic energy (TKE) is produced. As can be seen (Fig. 5 of the paper), TKE is equal to zero in the region in front of the cylinder, then is generated in narrow mixing layers above and below it and fastly moves downstream away from the cylinder, not having time and chance to suppress (via the turbulent viscosity related to TKE) the unsteady vortices coming from the surface of the body.

Finally, this set-up (zero free stream turbulence) is nothing to do with real situations in wind tunnels, water channels, and various cases in the nature. If one specifies a noticeable TKE level at the inflow, then all unsteadiness in the wake past the cylinder in RANS calculations may disappear due to the action of the large effective viscosity produced by the upstream TKE values!

Dear Sergey

I really appreciate your contribution, it is a further help to this discussion. What you linked is the classical framework presented in literature for RANS/URANS/LES and hybrid formulations. And is also what is generally accepted without further analyses.

However, what I wanted to stress in this post is the actual framework we are involved when considering the discrete equations we solve in practice. As I addressed, when you write the URANS and LES discrete equations they are totally equal, differences appearing only in the forcing terms, that is the turbulence models. That means that an implicit spatial filter is actually present in the URANS equations as well as is classically accepted for the LES ones. Consequently, we must focus on the expressions of the turbulence model. For LES we have a filter width, related to the grid size, that enters into the SGS model, exactly as you wrote in:

LES models do contain grid-related scales, and URANS models do not contain any quantities related to the numerical grid.

But the discrete equations we use in URANS are filtered (implicitly) in space, too! What about the theoretical justification for assessing that an URANS model must not contain such information? To give a proper justification we should analyse the form of the model that suppress the induced grid filtering? Is it a correct justification that the URANS model suppress any spatial scale separation? The ambiguity is that when we write the continuous equations, we assume that a spatial filtered is applied in LES and some time-averaging operator is applied in URANS and all seems clearly separated. But that remains only a formal framework, in practice we do not apply nothing and the equations are the same…

Now, considering the turbulence model effects you address after we could also cancel any effect of a turbulent viscosity. RANS solution is steady because we can just obtain that by statistically averaging a DNS solution. For statistically steady flow, the ensemble averaging has as same result as the time averaging. This RANS solution is exact, no need to introduce the concept of modelling and eddy viscosity. Conversely, we cannot determine an exact LES or URANS solutions from DNS. It is well known that filtering a-posteriori a DNS data is not sufficient to determine a LES solution, we have only a statically filtered field at some time. The same happens if we want to deduce a URANS solution from DNS, we should apply some time-averaging of finite width but that does not represent the real dynamic of the URANS solution. To determine a perfect LES or URANS model from DNS we should couple the residual terms determined from DNS to a real solution. In doing that we can adopt models such as scale similar one or deconvolution-based one, without introducing any eddy viscosity model. But I still cannot see how the grid length effect can be theoretically suppressed.

So please, try to discuss about some real URANS modelling and show me that no computational step sizes are needed.

Thanks

Filippo

Just to add that I invited the authors of the paper I posted to partecipate here and share their opinion but, so far, no reply were added.

This is one of the most exciting and interesting posts I have read here in Researchgate. Still, we have not seen an "agreeable" conclusion from the experts that have contributed to the post. I cannot express my eagerness to keep reading and learning from your contributions and I hope to see these comments, discussions soon in a formal evaluation in a paper, workshop or conference. It seems that this discussion has been delayed a lot in leading turbulence research groups, and yet, no clear conclusions are foreseeable. I would be surprise and glad to see the comments from the authors of the paper shared by professor Denaro.

Hi Julio,

unfortunately the authors did not express their valuable opinions.

I add a further article now that discuss further these issues. And again, the key appears in the adopted model that defines the real meaning of the resolved variables.

However, I agree, each one has a own idea about what is LES and what is URANS.

Regards

Filippo

I think I have to second Sergey N. Yakovenko, if I understood correctly his answer.

My current understanding is that, in their current implementation (which nonetheless might be wrong), standard URANS and RANS are models for fully tubulent flows, not capable of modeling transition.

If the (U)RANS inflow condition is such that the resulting turbulent viscosity is zero or has LES like values then, given the current implementation ambiguity, there is something wrong in one of the two.

That is, given again the implementation ambiguity and the fact that (U)RANS solutions should be fully turbulent, I would expect their predicted eddy viscosity to always be above a certain threshold to actually have meaning as fully turbulent solutions. Otherwise, we start having LES like behaviors for URANS and probably no convergence for RANS.

I don't know if such behavior is intended or not, correct or not, but agin it probably leads back to the original FIlippo question, I guess: is there something we should be doing in order to ensure that our URANS computation always produce an URANS solution? Maybe something along the lines of explicit filtering in LES?

Note, however, that implicitly filtered LES are still LES because a filter is actually there, despite implicit. This does not seem to be the case in URANS. Maybe the key to obtain this in URANS is just a sufficiently large time step? Not yet clear to me at the moment, considering that, in the RANS limit (an infinite time step) we still have non converging cases (even with the true Reynolds stress determined from DNS).

Let me add a few comments to further develop the informative answer of Paolo Lampitella :

I agree that RANS models are inappropriate for flow cases with transition to turbulence in some areas of the domain.

Originally, RANS approximations, proposed about fifty years ago to close the averaged NS equations, were based on the assumption that everywhere in the flow we have the developed turbulence state. Typically, this implies that in each point of the domain we have a quite long cascade of turbulent eddies where turbulent kinetic energy is generated by large-scale motions (low-wavenumber spectra range), then transferred from larger to smaller eddies (inertial spectra range) and finally dissipated by small-scale motions (dissipative range). And the inertial range should be quite long: then the energy-containing and dissipative ranges do not "feel" each other for the developed turbulence state. This also means that not only the Reynolds number based on freestream (inflow) velocity and body size, but also that based on the Taylor micro-scale of length are both quite large everywhere in the flow.

This is not the case for flows around bodies at not so high Reynolds numbers and very low (or zero) inflow turbulence levels. However, classical URANS models simply produced by addition of time-derivative terms to classical RANS models are still applied to these situations. It is not suprising that for such a case URANS tools may work essentially like DNS, ILES, LES as said earlier. However, URANS computations are still useful since they can visualize strong modes of large-scale (low-wavenumber) instability with less expense than in accurate DNS, ILES, LES runs.

Finally, we have a very challenging issue of inflow conditions. Incorrect inflow velocity and scalar distributions (caused usually by absence of accurate time-dependent data for instant fields) can eventually change and distort downstream distributions. And this influence often is even stronger than the effects of a particular turbulence model choice (RANS or LES or ILES). I have ever seen the situation (Article Numerical models of a far turbulent wake of an elongated bod...

Hi Paolo, hi Sergey

quite agree but it seems a strange joke...

if URANS is just adding the time derivative to RANS equations with a standard turbolenta viscosity there is no reason to do that. Such derivatives mean nothing physically, they MUST go to zero because they are nothing else that numerical residuals in the RANS equations.

There is nothing physical in the transient solution.

Conversely if the model is based on a certain time scale and is physically different from the standard RANS modelling we can try to discuss about what physically it reproduce.

It seems otherwise that URANS is some magic joke for lazy people that do not want to use LES (or poor people that has no computational power like me 😬)

Filippo, the best way to check your thought "that URANS is some magic joke" is to discuss this issue with researchers who have extensively used and compared URANS versus LES for various cases. Perhaps, you and they could meet each other at CFD or turbulence conferences, then generate the joint conclusions supported by physical arguments and share with us ;)

(I thought about this past ~1999 when first listened to URANS results at the conferences. Initially, was very shocked too by strange transition from RANS to URANS computations which suddenly give unsteady things, still with steady bc. Later, my opinion was transformed to believe in some help and evidence of URANS tools, when checked that DNS and LES can also give unsteady behaviors with steady bc, but indeed consume a lot of computing resources. Recall, DNS, ILES, LES are often insufficient to obtain accurate results due to strong grid-size requirements or/and because of the crucial inflow & initial data issue mentioned earlier.)

Chapter Performance of RANS, URANS and LES in the prediction of airf...

This is one more paper on "URANS vs. LES" with conlcusions:

"Although URANS also computes for transience, it fails to account for unsteadiness, and hence is not an appropriate replacement for LES.

...

URANS was unable to account for the fluctuations of the flow field, although solving for the transient solution, as it is limited to the externally induced fluctuations (i.e. periodic motion) which it failed to detect due to the relatively large turbulent boundary layer of δ/H = 5. Therefore, although URANS is comparatively cheaper than LES in terms of computational cost, it is not a suitable replacement for air pollution problems, or any other generic situation where small scale eddies are an integral part of the flow field development."

Many thanks Sergey, this conclusion is exactly coherent to the idea we discussed above about the URANS as a formulation based on the ensemble averaging. In such case, if the flow is statistically steady, the adding of the time derivative means physically nothing as it must go to zero to produce the RANS solution. Time averaging and ensemble averaging must converge to the same result. It makes physically sense only if an external time-dependent forcing is present.

However, this constraint can be overcome if we think to apply a different averaging, that is URANS seen as a result of a local time-averaging. This is the counterpart of the time-filtering in LES. But, as we never really apply this averaging, all the focus is shifted to the suitable model that defines it.

At this point I think that the literature has large lack in the analysis.

Just a joke I found on youtube to remember thinking about what we really solve :-)

https://www.youtube.com/watch?v=lkVfbJmWhk4

Interesting

All,

I would like to ask your opinion of a report we just finished on URANS. We are on the math side of CFD and coming from the LES side of turbulence modelling. So our intuition about what is important, what things mean etc is not well developed here. We also don't have a background of experience with RANS and URANS. While I'm confident in the long strings of inequalities, these are not the most important parts. I especially appreciate criticisms-it is how science progresses and it gives me things to think about next. The report is at:

Preprint On URANS Congruity with Time Averaging: Analytical laws sugg...

Thanks all, Bill

Naseem Ahmad

there are much more issues in URANS than only the lack of capability of simulating transition.

Naseem Ahmad

Hello.

Old post but I resume it to propose a question after a breif discussion with my collegue Paolo Lampitella

Assume a flow problem where turbulence is not statistically steady. Assume to adopt an ensemble averaging on the NSE and getting the time-dependent statistical equations.

It is suitable to assume that the ensemble averaging operator is idempotent without any ergodic assumption?