For as long (or rather, short) as I've been in the fields of turbulence research and computational fluid dynamics, I've been told that the Reynolds-averaged Navier-Stokes (RANS) approach of modeling and simulating turbulent flows is based on ensemble or time averaging of the Navier-Stokes equations. Here, the Reynolds stresses have to be modeled to close the resulting equations before they can be solved.
On the other hand, large-eddy simulations (LES) are (formally) based on (spatially) filtering the Navier-Stokes equations. In this case, the subgrid-scale or subfilter-scale stresses have to be modeled. In a commonly used approach to large-eddy simulations, the theoretical approach/description of filtering is just that: a theoretical formality that is not used in practice. A practical large-eddy simulation then consists in solving the Navier-Stokes equations on a grid that is too coarse to resolve all scales up to the Kolmogorov scale, but fine enough to still resolve the large scales. An extra forcing term (the subgrid-scale model) is added to model all the missing physics (for example, the dissipation of kinetic energy).
I wonder now how actual RANS simulations are performed, especially given the existence of hybrid RANS/LES methods? Does time or ensemble averaging ever play a role in an actual practical RANS simulation? I could imagine that time averaging implicitly plays a role if practical RANS simulations are like iterative methods that try to obtain a steady-state solution of the Navier-Stokes equations on a coarse grid. But then I don't understand how switching between RANS and LES modes, like in hybrid RANS/LES is possible. Or is the time/ensemble averaging of RANS just part of the theoretical description that is not used in practice, as is the case with filtering in the practical approach to LES I describe above? Is a practical RANS simulation then 'just' a very-coarse-grid simulation of the Navier-Stokes equations? (In which, for example, eddy viscosity or Reynolds stress models just serve as extra forcing terms to capture any missing physics?)
Hi Maurits,
I think it is safe to say that time/ensemble averaging is never performed in an actual RANS simulation, just like explicit filtering is not (or rarely) performed in practical large-eddy simulations. It is mostly the turbulence model that makes all the difference, and provides the variables in each formulation with the proper meaning.
In RANS, the eddy viscosity is such that the velocity field obtained from the numerical solution represents (a reasonable approximation of) the time/ensemble-averaged flow field of the problem under study. Since the expected velocity field is typically smooth and often two-dimensional, considerable simplifications are allowed, including 2D domains, coarse grids, and low-order/dissipative numerics.
Francesco
I will try to input an answer from my limited knowledge.
RANS simulations is not a very coarse grid approach of solving Navier Stokes Equations. Neither is DES (hybrid RANS/LES) because it has RANS in it for the near wall regions.
LES is a coarse grid approach for Navier stokes equations. Refine the grid fine enough to solve the smallest (kolmogrov scales) and you have DNS which is Direct Numerical Simulations in which NS equations are directly solved without any further assumptions. The only sources of errors are then from numerical discretization which can be controlled and quantified.
Basically RANS models simplify the Navier Stokes Equations by introducing the Eddy viscosity models which uses linear/Non-linear boussinesq assumptions. Linear Boussinsq assumption states that the Eddy viscosity terms are directly proportional to the strain rate magnitude of velocity.
TL;DR:
Let me add to/rephrase my original questions:
How is a RANS simulation performed in practice?
Which equations are solved? The Navier-Stokes equations? A simplified form of the Navier-Stokes equations (under assumptions of, for example, homogeneity)? The Reynolds-averaged Navier-Stokes equations in which the Reynolds stresses have been replaced by a model? The steady-state Navier-Stokes equations?
What kind of a grid is used? If homogeneous directions / directions of statistical stationarity are present, do we only need grid points in the other directions?
Does the original theoretical formalism involving Reynolds/time/ensemble averaging play a role in practical simulations or not? That is, does one ever perform time averaging explicitly in the numerical solution procedure? Is time stepping used or does one look for steady-state solutions?
If no homogeneous directions are present, how is a RANS simulation different from a (coarse-grid) numerical solution procedure of the Navier-Stokes equations in which an extra forcing term is added to model/replace the turbulent stresses (as is a common approach in LES)?
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Dear Mohamed,
Thanks for your answer and your nice way of explaining large-eddy simulations. It supports the view that I have (and use in my practical simulations) that in large-eddy simulations one basically solves the Navier-Stokes equations on a coarse grid.
Thanks also for your clear statement that RANS simulations are not coarse-grid simulations of the Navier-Stokes equations. I still do not understand, however, how a practical RANS (or hybrid RANS/LES) simulation is set up?
You remark that RANS models simplify the Navier-Stokes equations by replacing the Reynolds stresses by eddy viscosity models or nonlinear models (or, of course, a host of other models). The very same can, however, be said of large-eddy simulation: extra forcing terms (subgrid-scale models) that can be of eddy viscosity type, nonlinear type, etc., are used to replace the turbulent stresses in the filtered Navier-Stokes equations, in essence leading to the Navier-Stokes equations with an additional forcing term.
So, let me rephrase my questions after describing in a bit more detail how large-eddy simulations can be performed in practice: I use large-eddy simulation as a coarse-grid numerical solution procedure of the Navier-Stokes equations, in which extra forcing terms are used to describe any physics that is missing/unresolved (originally contained in the turbulent stresses that one obtains in the formal description of LES involving filtering). Here the grid can be coarse (otherwise, indeed, we do a DNS), but still has to be fine enough to resolve large scales of motion. From this procedure one obtains a solution (velocity field, pressure) as a function of three-dimensional space (in discrete points, of course). This solution is also time dependent, since the numerical solution procedure involves time stepping.
In short, the result of a large-eddy simulation is a time-dependent solution of three (discrete) spatial coordinates. It is hoped that the extra forcing terms (the subgrid-scale models) do such a good job, that this solution is representative of the large scales of motion of a full-scale direct numerical simulation of the Navier-Stokes equations.
How does a practical RANS simulation work in comparison to the above description? Which equations are solved? What kind of a grid is used? Is time stepping used or does one look for steady-state solutions? Is the resulting solution a function of all spatial coordinates?
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As an example, let's consider the simulation of a plane-channel flow. This flow can be expected to be homogeneous / statistically stationary in the streamwise (x) and spanwise (z) directions. We could, therefore, decide to include averaging over these homogeneous directions in the Reynolds averaging procedure. Thus, the Reynolds-averaged Navier-Stokes equations can be simplified substantially: all derivatives with respect to x and z can be removed. (The solution will also have to be constant in those directions.) I could imagine that performing a RANS simulations of this test case then amounts to (i) replacing the Reynolds stresses by a model term and (ii) solving the resulting modeled and simplified Reynolds-averaged Navier-Stokes equations. These now only depend on one spatial coordinate, the wall-normal coordinate (y). If Reynolds averaging also includes a time average, we could easily find a steady-state solution of these simplified equations, which, in the end, are only a function of y. If time averaging is not part of the Reynolds averaging process, I guess we would obtain a URANS solution, a function of the wall-normal coordinate (y) and time (t).
If the above description of a RANS simulation of a plane-channel flow is correct, I must conclude that all that is different with respect to the LES is that we solve a simplified variant of the Navier-Stokes equations (given homogeneous directions / directions of statistical stationarity).
I must, however, have made a mistake somewhere in my above description, because what will happen if homogeneous directions are not present? In that case the Reynolds-averaged Navier-Stokes equations do not simplify (apart from the replacement of the Reynolds stresses by a model). As a consequence, we would have to find a numerical procedure that leads to a solution that depends on all three spatial coordinates (and possibly time for a URANS?). I don't see, however, how RANS would save us any computational effort in this case unless (i) we are only interested in steady-state solutions, or (ii) a coarse grid is used.
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After my lengthy (original) question and this additional lengthy discussion, I hope someone can clarify how a RANS simulation is performed in practice and if the original theoretical formalism involving Reynolds/time/ensemble averaging plays a role in it or not? Thanks!
Hi Maurits,
I think it is safe to say that time/ensemble averaging is never performed in an actual RANS simulation, just like explicit filtering is not (or rarely) performed in practical large-eddy simulations. It is mostly the turbulence model that makes all the difference, and provides the variables in each formulation with the proper meaning.
In RANS, the eddy viscosity is such that the velocity field obtained from the numerical solution represents (a reasonable approximation of) the time/ensemble-averaged flow field of the problem under study. Since the expected velocity field is typically smooth and often two-dimensional, considerable simplifications are allowed, including 2D domains, coarse grids, and low-order/dissipative numerics.
Francesco
I will try to give my idea.
RANS: The statistically averaging is implied in the form of the equations. You get a steady state formulation and the model must take into account that fact that all the characteristic scales are involved by the averaging. In practice, you solve the RANS equations as they were for steady laminar but adding the model closure based on physical assumption that are problem dependendent.
A variation that takes into account a time evolution of the statistically averaged fields is the Unsteady RANS (URANS). It could be questionable the meaning of such unsteady term added to the RANS equations. But this is another hystory.
LES: if you use an implicit filtering approach, that is the filtering is due to the global effect of the computational grid along with the discretization of the operators, you will see a form of the filterd equations very similar to that of the URANS equations. However, the meaning of the resolved filtered variable is very different and the SGS model acts on the basis of very different physical assumptions.
Last observation: the accuracy order of the discretization is relevant in LES but much less in RANS/URANS wherein the closure model has some order of magnitude greater than the local truncation error.
It is all about Eddies. How?
DNS: solves NS equations for all Eddies (Large and small), hence no turbulence modeling needed. But this needs very fine mesh (nodes number ~ Re^(9/4)) and CPU-time ~ Re ^ 3.
LES: solves NS equations for some Large Eddies and model small ones. So, Less number of mesh and Less time, but less accuracy as well.
RANS: solves NS equations for only largest Eddies and model all other Eddies. Then, coarse mesh compared to LES and DNS but with special conditions among RANS turbulence models.
DES: uses RANS near walls and LES away wall.
About RANS averaging: actually, Eddy-Viscosity models and Non-Linear Eddy Viscosity models, all seek one target which is; determination of Turbulent Eddy Viscosity (Mu_t) to determine Reynolds/Favre stress tensor.
Moreover, Reynolds stress model (RSM) solves Reynolds stress transport equation to find six independent variables of Reynolds stress tensor.
Dear Omer,
RANS does not work as you wrote, in such formulation there is no distinction between "largest" and "other" characteristic flow lenght. The statistical averaging implies that the whole flow scales are involved and therefore the model depends on the features of all flow lengths. This is the reason why RANS modelling is so strognly dependent on the BC.s that characterize the flow problem.
Dear Filippo,
Greetings!
Boussinesq (1877) observed that the momentum transfer in a turbulent flow is dominated by the mixing caused by large energetic turbulent eddies. Then he proposed his Eddy Viscosity hypothesis (linearly dependent) according to this observation.
Regards
That has nothing to do with the concept of statistical Reynolds averaging but was an empyirical consideration about the macroscopic effect of turbulence on the mixing.
To understand the cocept of statistical averaging, consider the 2d flow of a laminar vortex shedding behing a cylinder. The only frequency you get is due to the largest vortical structure alterning the sign in the vorticity. Now immagine to perform a statistical averaging of such solution by means of a temporal averaging over a very large period. The resulting solution will be steady with two counterotating small structures and you get no longer characteristic frequency of the large eddy. This is the equivalent of the RANS solution.
I have thought a way to connect LES and RANS through a Langevin equation.
This can be used for example in order to create boundary conditions for LES.
First you run a RANS simulation and you store the results of the velocity mean and variance, turbulent kinetic energy (k) and dissipation (ε). The two latest variables are used to calculate the autocorrelation time scale e.g. for the atmospheric surface layer using the equation T=0.5*k/ε.
Then using the RANS output you create velocity time series using the Langevin equation (see attached paper) with specific turbulent characteristics e.g. autocorrelation time scale and variance. I use Gaussian distribution for the random parameter and I am going to replace it with a beta distribution based on one of my publications in Boundary Layer Meteorology.
Finally you use these time series for LES. I have used this method in a real industrial accident and it worked (attached paper). This peridod I further validate the new methodology.
So, let me try to go back a little bit back to the original question, and try to keep it very practical practical. I mean to represent 'the industry' here, not the tweaking or specials that might be going on in Universities:
For most RANS formulations (Except e.g. RSM or EARSM) one starts with the Navier-Stokes equations after assuming Newtonian fluid, so you're basically left with the viscosity mu times the shear stress. Then most RANS turbulence models do nothing but just replacing mu by mu + mu_t, the eddy viscosity. That's it... eddy viscosity is all that communicates the influence of turbulence to your momentum transport (let's not bother with energy equations or compressibility right now, I usually deal with water). Of course you need the (usually) one or two turbulence equations to model mu_t, which is a function of x,y,z and t (time). Because of this most RANS models imply homogeneous turbulence.
So, theory can tell you what these two turbulence equations should look like (an important step, but we'll skip over it right now). And there is sometimes some cheating going on there as well, since in two equation models sometimes the form of the dissipation equation is taken similar as the kinetic energy equation (like in the k-epsilon model). So, at this point you are left with two equations modelling one variable mu_t using around (depending on the model) five empirical/modelling constants in these equations. This is more or less the point where theory stops and datafitting begins. Although one of the constants can be related to the dissipation of turbulence behind a grid and more of these kind of relations may exists, what you are basically doing at this point is searching which values for the constants give the best representation of the data (a large bunch of measurements). And mind you, a good fit is if the averaged velocities are predicted well by the model, not (or to much lesser extent) whether the turbulence quantities are predicted well.
Now, I'm taking a great bunch of short cuts here. And people dealing with turbulence modelling might be a bit annoyed by me reducing this 'black art' of turbulence modelling to data fitting. Now, this data fitting can make great use of theory, but in the end this is what you are doing in my opinion.
The same practical approach is used for gridding. We're dealing with practical problems for which usually the complexity of the geometry dictates to a great extent the types and shapes of grids that you can use. Structured grids are most of the time out of the question and we'll use whatever works (grid quality is important!). Now, you do need to be very careful to get a grid independent solution, but other than that, use what works.
Another thing to remember is that by far most of the problems that RANS successfully deals with are steady problems (i.e. the solution to the RANS + turbulence model equations are steady) and therefore ensemble averaging is of no use (or very easy, depending on your point of view :) ). Also, I think the solution to unsteady problems is usually rubbish (by which I specifically mean unsteadiness due to separated flow or unsteady flow features like free shear flow. Unsteadiness due to moving geometry can be useful). Maybe integrated quantities like forces are of some use, but the field values of the velocity and turbulence are generally not. For this you really need to go to other turbulence modelling (DES/LES, or maybe PANS, SRS or the like). In other words: RANS models work well for aerodynamic shapes, outside of that much less so. Ok, blunt bodies for which unsteadiness of the flow is not that important (i.e. the RANS solution is more or less steady) usually do reasonably well also.
Lastly I would like to remark that RANS usually introduces a LOT of eddy viscosity. This usually kills a a lot of the unsteadiness of the problem. More advanced turbulence models usually lead to lower eddy viscosities (specifically PANS or SRS models) which then allows for more dynamics.
Dear Arjan
you are right about the empirical and practical aspects of RANS, the experience of the user is fundamental in setting the suitable model parameters.
I want just stress the concept that RANS MUST be steady by definition of Reynolds averaging over an infinite number of samples, invoking ergodicity and using the time averaging for an infinite time.
When we switch the steady equation to unsteady equation, the time derivative changes the meaning of the solution. When an external force is present, you can consider the field at a certain time as the result of the ensemble averaging where time is a parameter. Then you proceed for other time steps. The classical example is the flow in a cylincer with a moving piston. The cranke angle stands for the time.
But a different view could be introduced for the unsteady equations when a time filtering is supposed. That is a way to see the URANS equation as LES wherein only a time-filtering is used and the closure models has several aspects to take into account, the RANS modelling in space and the SGS modelling in time. I see such framework considered only in few studies. But that could be analysed and developed to better taking into account the hybrid approaches. For example in the Embedded LES.
I have to agree with Fillipo about omer's explanation on RANS. it is quite wrong.
About Arjan's reply: You mentioned that RANS eddy viscosity models are a function of X,Y,Z and time. this is also not true. RANS eddy visocisty models are a function of (U and time). More specifically a function of strain rate magnitude and how it changes over time.
I don't particularly agree with Arjan that URANS produces rubbish results as well. In many cases RANS with unsteady time marching can produce good results with correct strouhal, time periods and force coefficients.
Neither do I agree with the fact Arjan mentioned that RANS always overestimates eddy viscosity. No. Sometimes it is underestimated and sometime it is overestimated.
Hi Mohamed,
" You mentioned that RANS eddy viscosity models are a function of X,Y,Z and time. this is also not true. RANS eddy visocisty models are a function of (U and time) "
Yes, and U is a function of X,Y,Z... All I was saying is that mu_t is not a constant, but a variable for which you need (often) one or two equations to solve. I'm not suggesting that the equation(s) determining mu_t somehow directly depend on X,Y,Z...
For URANS (which is a funny name, the general RANS equations are already time dependent, but let's go with it) my experience is that once the flow has separated or contains a free shear layer, the dynamics of the flow can be quite inaccurate. So what I'm saying is that if you need to know about velocity fluctuations and/or turbulence quantities in these cases, you need a more sophisticated model.
About the eddy viscosity, I didn't say that "RANS always overestimates eddy viscosity", I said that RANS usually introduces a lot of eddy viscosity. Usually is not always but I think I'm accurate when I say that RANS introduces a lot of eddy viscosity in turbulent regions which kills part of the dynamics. This generally becomes less with more sophisticated models.
Arjan,
" which is a funny name, the general RANS equations are already time dependent, but let's go with it "
assume the definition of RANS variable:
= lim N->+Inf (1/N) sum i=1,..,N f(x,t)
and now consider this definition at time t+dt. What should change in the average from (t) to (t+dt) if you are averaging over all possible infinite realizations? Nothing change in time, except the case you have an external forcing (like the example of the piston moving in a cylinder). You can see the book of Frisch when he discusses about ergodicity.
URANS is not a funny name but a different concept in the formulation.
It is not easy to say wrong to that interpretation about RANS. let's see!
(1) In Mathematics:
Let's consider incompressible flow (Density=constant) without energy equation, So we'll have only 1- continuity Equation 2- momentum equations (x,y,z) which are called NS equations.
Now, if we apply either the time averaging or the ensemble averaging to NS equations the resulting equations are called Reynolds - Averaged NS equations (RANS) which are FORMALLY IDENTICAL to the original NS equations with the exception of the additional term of Reynolds - stress tensor (RST)
Therefore, in practice, all RANS models job is to APPROXIMATE/PREDICT the RST and ADD it to the ORIGINAL NS equations. So, solving NS equations for mean values and add RST to momentum equations.
EVMs and Non-linear EVMs do this job by replacing RST by and similar equation for Nonlinear EVMs (S can be calculated), So, ALL RANS turbulence models seek to calculate mu_t ONLY.
HOW? EVMs assume that the kinematic turbulent viscosity can be expressed as a product of a turbulent velocity SCALE (v) and turbulent LENGTH SCALE (l), which are approximated through Turbulence models (one or two equations).
So, if the turbulence model capable to predict the accurate LENGTH SCALE for each control volume then the simulation is good enough if the control volume is fine enough. Which means that the LENGTH SCALE is affecting the Mu_t then RST then RANS approach. This supports my interpretation about RANS.
(2) In physics:
The Boussinesq hypothesis of Eddy Viscosity is based on the observation that the momentum transfer in a turbulent flow is dominated by the mixing caused by LARG ENERGETIC turbulent EDDIES. (check Re[1]).
It is easy to notice that if you compared contours of RANS and LES or DNS. you will see that RANS is not able to visualize small eddies. Even you compare LES with DNS you will see LES ignores very small Eddies.
(It is all about EDDIES)
Re [1] J. Blazek, Computational Fluid Dynamics: Principles and Applications, 2nd Edition, 2005
I do agree with Arjan about URANS.
There are three forms of Reynolds averaging :
1- Time averaging (statisticlly steady turbulence) - the mean value doesn't vary in time but only in space.
2- Spatial averaging - mean value uniform in space and vary in time.
3- Ensemble averaging (unsteady simulation)- mean value remains function of time and space.
But if the density is not constant we use :
Density weighted or Favre decomposition (Favre averaging)
Usually, for compressible flow, it is convenient to employ Reynolds averaging for density and pressure and Favre averaging for other variables which is called Favre-and Reynolds averaged NS equations.
Omer,
as reported in the textbook of Wilcox the three definitions of averaging do converge each other when ergodicity is assumed. Therefore you get steady equation for each of the three definitions. We can debate about the discrepancy in the time and ensemble averaging for large but finite T in the theory of large deviation. This is explained in the textbook of Frisch.
I give you a physical example of a turbulent problem, well-known in the CFD literature: the 3D flow in a plane channel problem. Assuming you are doing a DNS for this problem (but we can suppose a laboratory experiment is ideally well done) you get a 3D and time-dependent solution. Ideally, you can also generate infinite realizations of this DNS solution by repeating the simulation for different initial condition.
Now, after a statistical energy equuilibrium is reached, try to do both the time-averaging and the ensemble averaging. Asymptotically, they go towards a 2D steady solution that is the RANS solution for this problem. You could think about a "fictitious" time-dependent solution if the averaging is performed over a finite T such that large oscillations are still present. You could also denote URANS that solution but it is a result of an incomplete statistical sampling.
Conversely, if in the same problem an external forcing is added, for example a periodic pressure gradient, you could introduce the ensemble averaging that will produce an unsteady term in the equation.
Obviously, you should consider that RANS and URANS would require some difference in the model assumptions.
Maurits- You are asking questions that people in the field have agreed to ignore: for ensemble averages, there is no random variable introduced and no modification in the model for putting it with different PDFs in different places and for URANS the time averaging window does not influence the model either. In my discussion with industry where RANS is used a lot, they will essentially say RANS is feasible for them and they know what information from it to believe. They are also solving similar problems again and again so know how to calibrate the models for their flows. In my opinion, increasing computer power and the need to do UQ means that we will soon enough be solving for ensembles of solutions from which means and fluctuations can be calculated directly rather than modelled (and then used to determine nu_T). I am surprised that for URANS no one (to my knowledge) has tried testing something like Germano's dynamic model where time averages over a double window are used to extrapolate and predict the leading order time fluctuation and from this determine the turbulent viscosity. This would give a URANS that is self-calibrating and changes with the time window. .... One good thing about turbulence modelling is that (paraphrasing Laplace) what we dont know is so big and what we know is so small that there will be much to do for a long time forward.
William J. Layton
Hello, there are many controversials about what URANS actually means, some time ago I opened a specific post on RG to discuss that. The idea of defining a time-based Germano identity is possible as URANS can be seen in terms of a time-filtered LES. Papers in literature exist that propose such an approach. What is quite difficult to extend is the concept of a proper test-filtering in time, that is define a specific time-width for which we can assume a location in an inertial sub-range. I think that this topic is suitable to further studies but my idea is that you go into the evidence that a spatial filtering (implicitly induced by the discretization) acts along the time-filtering.
Thanks Filippo!-I'll search for the papers with interest. Is there an interpretation of URANS other than coming from a fixed time window average? It's not what I've spent time doing and my knowledge is based on arguing (in a friendly spirit) with people in industry committed to it. An unrelated question: Can you recommend a simple-as-possible test problem where RANS/URANS gives O(1) wrong results? --Bill
Hi William, I can address some papers linked into the discussion I opened
https://www.researchgate.net/post/URANS_what_is_the_meaning_for_statistically_steady_flows_and_what_compared_to_LES
A unique framework between URANS and time filtering LES would be possible but, to the best of my knowledge, has never been formally elaborated. In my opinion the key is that the filter or the time averaging are generally never explicitly applied and all the differences between the formulations are intrinsically due to the used model.
In the current implementation practice, I think, it is safe to say that RANS could be termed, in comparison to LES, as IMPLICIT RANS, meaning that, just like in implicit LES, the operator acting on the original equations (i.e., NS) to obtain the RANS ones is only implied by the procedure used in the overall numerical approach.
As a consequence, in both cases, implicit RANS and LES, all the information and burden required by the specific formulation (RANS or LES) is necessarily carried by the specific turbulence model in use, i.e., the forcing we introduce in the equations.
The main difference between a typical implicit RANS and LES forcing/model is in their dependence from the very implicit operator they are trying to follow/determine.
Implicit LES is quite clear on this, as each model depends from a filter width that is, in a way or another, linked to the grid, which is recognized as the primary implicit filter together with the relative numerics acting on it.
In implicit RANS, and this is one of the two main things that separate it from implicit LES, it is not anymore dependent from the grid or any other filter width or sort of.
The point is that, actually, in implicit RANS models, there is no actual dependence on anything related to the average underlying it (I'll get back to the ensemble/time average issue in a second). Basically, every implicit RANS modern model depends from one or more equations that, in turn, only depends from their bc. As a result, if there is any dependence of the actual model from an underlying average (in the same way implicit LES depends from the filter width) it is either in the meaning, implicit again, of the variables used and the values, explicit, you assign to them at the boundary, or in the specific form of such added equations. I think it is fair to say that, basically, the only explicit, practical way to connect an implicit RANS model to a certain average is by giving specific bc values for the variables used by the model (yet, I am not aware of any study in this regard).
Now, the second main thing that separates implicit RANS from implicit LES is that, of course, implicit RANS implies a steady set of equations. This, in theory, would not be really necessary if we adopt the ensemble averaging point of view, but we have to confront ourselves with the actual practice and nomenclature (to which the original question is related). In such practice we have a distinct place for Unsteady RANS (URANS), to which I'll come in a second.
What I want to highlight here is that this is kind of an unicum situation or, put differently, one of those gray areas in implicit RANS which are just starting to be explored in the literature. Basically, we both force the implicit RANS behavior trough the model (and the bc values of its underlying variables) and trough the fact that, in practice, we try to achieve a solution with a steady algorithm (which is legit, we want to solve a set of equations which are steady, by definition if we imply a time average). Unfortunately reality kicks in here, and what you find is that not all the cases where you try to achieve a steady RANS solution are actually workable. Roughly speaking, some cases will simply not converge with any given model (yet, nobody explored different turbulent bcs in those cases). For example, for strongly separating flows, we can define an average (and it will be time independent if the bc are so), but no model is capable to give you back that steady solution. What is more disturbing is that, even if you perform the same average on a DNS and try to enforce the relative Reynolds stresses in an implicit RANS of the same case, that wouldn't probably converge neither, which is what the recent research has found while exploring AI based formulations of RANS models.
There it comes URANS. From the implementer point of view, let me say it, it just makes sense. You have every piece in place to make it available, why not? Also, it just fits the same framework as long as you imply that the underlying average is of ensemble type and not a time average, as in practice implicit RANS implementations actually imply (i.e., the steady algorithm).
Nonetheless, Implicit URANS (that would be the correct nomenclature here) is still another beast. You have the same model as in implicit RANS, but now, formally, you are also applying an implicit time filter trough the numerical time advancement, which is largely recognized as another form of LES. Formally, implicit URANS makes sense if the implied average is of an ensemble type AND, in addition, if the implicit time filter does not affect any resolved scale (i.e., there is no resolved energy at the scales affected by the implicit time filter).
Now, and here I come to the questions recently put forward in this thread, I think there is nothing bad, per se, in this definition of implicit URANS. It would work like a charme in all the cases where the bc are slowly varying in time and no spurious scales (i.e., interfering with the time integration/filter) are produced in the process.
The problem is that you have no mechanism to control such interference, and when it happens you may notice it by experience or by filtering, but there is nothing you can do except reverting to LES.
Moreover, and this is even more disturbing, the unsteady term in implicit URANS can, by its very nature, work as a trash bin for anything that doesn't work at the underlying implicit RANS level. That is, by definition, the unsteady term collects anything that, at that time level, turns up being unbalanced. So you can't know if the spurious scales are produced by genuine non linear mechanisms from previously resolved and admitted scales or, say, by the same implicit RANS model that didn't work at the steady level as well.
As a side note, the same uncertainty on the nature of the unsteady term would be present in implicit (and explicit, for that matter) LES as well, but there it is well recognized as a possible source of error in the a priori comparisons, and as a consequence in a posteriori dynamics.
Paolo I definitely agree, there are lacks in the proper definition of bcs for RANS/URANS (as well as for LES) and a theoretical unique framework for the implicit approach to URANS/LES would be required. I recognize that such a work is not of industrial appeal, neverthless it seems to be not appealing in the academic community, too.
Even if I don't work for a major industry player, I can give you my point of view as implementer and code mantainer. Which, by the way, I think it is common to other areas of CFD. The point of view is that, unless the company has a specific competence on the matter (and typically doesn't, especially on turbulence, as you would end up with very large teams to cover everything, nor such experts are typically willing to work in industry), and there is a clear possible outcome from the effort, there will be no point in doing anything than simply look at the best things the academy has to offer. And honestly, I don't see how this could be even be of interest in most engineering departments today, understanding what is that we currently don't understand of seemingly working models. Quite a cryptic scenario here
Paolo, the issue is quite simpl: industry uses URANS without understanding that, from a practical point, are the same equations they would use for performing LES. Do they understand that?
I agree that the code does not know if it is implemented for URANS or LES, assuming an eddy-viscosity type model is used. Then, the problem is all about the simulation uncertainty and modelling error. In terms of URANS, my current understanding about its limitations is:
So, one can always manipulate URANS settings to `fit’ a particular experiment, but both researchers and practitioners should be careful in trusting URANS for `3D, unsteady’ prediction.
“These days it is common to see a complicated flow field, predicted with all the right general features and displayed in glorious detail that looks like the real thing. Results viewed in this way take on an air of authority out of proportion to their accuracy. In this regard, modern CFD is a very seductive thing”-Doug McLean.
Serious CFDers never trusts their own results unless there are sufficient evidences to prove the correctness, robustness and repeatability.
Dear Wen Wu
I agree, your conclusion is a correct warning to newcomer in CFD.
Let me comment topic 3): implicit time window width in URANS is somehow a counterpart of the implicit spatial filter width in LES. We can suppose that the time window is an implicit result of the action of the time discretization and the time scale assumed into the turbulence model.
Much more details were discussed in my previous post https://www.researchgate.net/post/URANS_what_is_the_meaning_for_statistically_steady_flows_and_what_compared_to_LES
Dear Filippo Maria Denaro
Thanks for the comments. I think the implicit time window in URANS is different from from the implicit spatial filter in LES. In the later case, the filter is at least comparable to the local grid spacing. So we have control over the resolution (model contribution) of the LES. In URANS, we know nothing about the temporal resolution and have no way to tune it. My point, then, is that even if URANS results could show small scale structures, we don't know which ones are of the `resolved' range. In other words, one can think URANS as phase averaging, for which the smallest time scale of the `resolved' periodicity is non-trivial. This is my understanding.
Wen Wu
actually, the time window would be implied by the characteristic time introduced in the model. That is the counterpart of implicit-filter LES I mentioned. The URANS model should contain the infos for the window. That characterizes the real difference from the statistical RANS formulation. Actually, this aspect is not formalized in a theoretical way.
Filippo Maria Denaro Thanks for the comment! The characteristic time scale introduced in the model depends on the solution. It is hard to tell a time scale such as k/epsilon is chicken or egg. So again, I believe one have no control over the URANS uncertainty especially in the temporal point of view.
Dear Wen Wu
yes, I agree about your comment I want just to highlight that URANS seems not really formalized in a theoretical way, the time window being not practically applied on the equation (see the post I opened some months ago) as same as happens in the implicit-based filtering LES. As a conclusion, the closure model must contain the infos about the characteristic scale. In LES this is theoretically developed in quite different ways but in URANS is not. Furthermore, the dynamic time window could be deduced extending the concept of dynamic formulation in time. Again, many idea were described in my post, I would be glad if you give there some commens.
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