In a really turbulent regime, instantaneous flow fields will not be the same on the long run. At least round off errors will ensure some differences will be present and the underlying turbulent dynamics will amplify them. In many cases, random perturbations are added to the initial datum or forcing in order to accelerate this process. The simulation output is then ideally considered as one realization of a stochastic field and a Monte Carlo procedure is employed to compare statistics. Since this might be very CPU time consuming, often an implicit ergodic hypothesis is assumed and one compares space and time averages from DNS data. DNS is perfectly meaningful (provided the necessary resolution has been used by the numerical method applied) if the results are compared this way. What is not meaningful is comparing snapshots of DNS simulations,unless one is comparing general features only (e.g. scale of the typical vortices).

Thank you very much, Dr. Bonaventura,  your reply enhanced my understand. If the boundary and initial conditions are fixed (I mean the initial artificial perturbations also the same) for two case, the flow field are certain and obviously the same for these two case for any time if we use DNS. But if we use analytic methods (provided the NS equations can be solved analytically) under the same conditions, the solutions may be  non-unique, because turbulence is chaotic.  Do you think my thought is right?

I agree with the comments of Luca, however, we should consider also the information provided in the paper by Javier Jimenez

"Computing high-Reynolds-number turbulence: will simulations ever replace experiments? "

The text was published in the Journal of Turbulence in 2003 and it is available on the web; if you enter the title as the keyword in your google search you should be able to obtain the PDFs file (please let me know if you have problem with downloading the text)

The answer to your question is provided by eqs. 1-5 and the data in table I in the Jimenez (2003).

All the best,

Janusz

Thank you, Prof. Pudykiewicz. I have downloaded this paper and will read it carefully. 

I believe that analytic solutions in a turbulent regime are out of the question. I do not know any example and I doubt there exist any. Therefore the discussion restricts to laboratory experiments (as Janusz points out) and numerical experiments. Also in numerical experiments, in the turbulent regime non uniqueness of the solution can easily arise due to the (unpredictable) behaviour of round off errors. DNS, when correctly carried out, provides an approximation of one realization of the state of the turbulent fluid. Depending on how you use this information you may be drawing correct or incorrect conclusions from your DNS.

According to a summary presented by U. Fresh in the mini-monograph "Turbulence; The legacy of A. N. Kolmogorov", the road leading to understanding of turbulent flows is not unique. We can choose at least four identified climbing routes to reach the Navier-Stokes peak. A concise graphical summary is presented in the attached figure from the above mentioned book. According to the vision advocated in this figure, the numerical route is a school to take before attempting new climbing routes to the Navier-Stokes peak.

First, turbulence is not a pure chaotic phoenomenon... if you see a turbulence signal in the wavenumber space it is not a spectrum of white rumour addressing a random field....That is exactly why turbulence is still so challenging....

Second, DNS is not a model but it is simply any kind of discretization of the Navier-Stokes equations as they are, without adding any empirical  model on the physics. The key to call DNS is in the grid resolution, you must have a size h so small to resolve up to the smallest charactereistic scales of tutbulence. Thus, do not make confusion between DNS and no-model LES, this latter being actually an underresolved DNS.

Third, DNS (and also LES) do not aim to get comparable instantaneous fields between different simulations. Even the same code that run on different CPUs will produce different fields at the same time (for long integration). Turbulence has some deterministic features that we are able to reproduce and compare in statistical sense.

Having infinitely fine grids would NOT let cfd get real turbulent solutions dueto numerical scheme oscillations and dissipation. Even not considering round-off errors. How could one separate numerical and physical oscillations and dissipation? Numerical people love to say that high-order schemes should be used and they should produce better results but above question still holds.

Fundamental concept underlying numerical simulation is that of the convergence. Using convergent numerical schemes assures that numerical solutions amount to analytic results on "infinitely fine grids" (quoting Anton Noev). To be precise, this also requires exact arithmetic free of round-off errors. So, in principle, DNS can simulate turbulent flows.

I agree, DNS can simulate all the statistical features of turbulence. Of course, provided that the original set of PDE.s equations is the correct mathematical model for that specifici flow problem.

By definition, in a DNS the grid has still a finite size (Nyquist characteristic lenght) but, in terms of the flow problem that extends the lenght scale up to Kolmogorov lenght, it can be considered as the local truncation error vanishes.

well said!

dear Piotr

as far as i know numerical simulation of supersonic flows leads to oscillation of solution where local gradients are high. without adding artificial viscosity one can not get reasonable solution. so how to answer my previous questions? and how to analytically or somehow prove convergence for non linear scheme?

dear Filippo, as far as I know, Kolmogorov scale applies only to isotropic subsonic turbulence on flat plate. 3-d flows could easily have anisotropic turbulence like on attachment line of swept wing.

Dear Anton

1) oscillations in solutions with shock (singularity in inviscid flows) have a mathematical difference: the inviscid solution is not regular therefore you cannot define the classical "convergence" as the local truncation error cannot be defined across a shock where derivatives do not exist. Artificial viscosity is just an escamotage to have the original Euler equation modified in terms of a perturbation model that makes the solution regular. However this is a quite old framework, more recent numerical methods are adopted for the Euler equations.

2) No, Kolmogorov scale definition is not limited only to flat plate. And the flat plate is not an isotropic case of turbulence. In simple word, the Kolmogorov scale for a flow can be defined as the lenght for which the viscosity of the fluid and the characteristic velocity at that scale make the Reynolds number equal to unity.

dear Filippo

could you please provide some links to the best dns calculations?

dear Anton,

consider typing on google scholar "direct numerical simulation of turbulent flows" or something in this spirit.

In the case of compressible turbulence, I think one must consider whether the (compressible) Navier--Stokes equations are a correct model. It is well-known that NS equations do not correctly predict the measured width or shape of shocks, also a high Reynolds number phenomena. The issue is the lack of local thermodynamic equilibrium, upon which the derivation of NS from the more fundamental Boltzmann equation is based.

Dear Piotr, I encounter dns papers or presentations from time to time. As a rule all of them end up like "to sorry we did not resolve all eddies, we need more computational resources" or "in general there is good agreement with experiment. but only in general, excluding some points". I thought if you rely so much on dns you must know some milestone paper that removes all doubts about dns potential. Let it be incompressible flow with moderate Re number. Scholar gives tons of papers I would die to go through them.

Not an expert of experimental measurements, at the best of my knowledge in the shock wave field I know that the NS equations are a quite adequate model for moderate intensity shocks, that is until chemical reactions, ionization and non-equilibrium effects become relevant (a field of hypersonic flows). However, also for moderate shocks its characteristic width is so small ( of O(10^-6)-O(10^7) meters) that is always much more below the Kolmogorov lenght scale. Thus, as a general definition of DNS, we are not able to perform a real DNS for analysing the shock-turbulence interaction.

But maybe in the specific literature there new progresses I am not aware of ...

Dear Anton

an interesting problem is the wall bounded turbulence. Search for the paper of Jimenez & al. that performed a DNS. Have a look here https://www.google.it/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwjO7cudusbZAhVBa1AKHRRnBVUQFggtMAA&url=http%3A%2F%2Fwww.mdpi.com%2F2079-3197%2F4%2F1%2F13%2Fpdf&usg=AOvVaw2wx8mEiOHCs18H2FvQ5WsZ

Then, have also a look here http://turbulence.pha.jhu.edu/datasets.aspx

I suggest to see also the realizations performed at the CTR

"quite adequate" and "moderate intensity" are qualitative terms. Look at: Schmidt, 1969: ``Electron beam density measurements in shock waves

in argon," J. Fluid Mech. {39}, 361--373, especially figure 10. You will see

that by Mach 2, nature's shocks are already twice as wide as NS predicts

(ionization doesn't become important until about Mach 10). Measurements and DSMC simulations (of Boltzmann eq.) agree quite well, but show significant disagreements with NS in terms of shock shape. Even in 1922, Becker showed (theoretically) that NS would not be adequate for shocks. If this is a result of high Reynolds number (meaning that in the Boltzmann eq. it is the advective terms rather than the collision integral that dominates), then at the very least it is incumbent on the modeler to do more than simply assert DNS is "quite adequate."

Ok, but when we use the term DNS we focus only on the fact that the discretization of the spatial domain is so fine to resolve all the characteristic turbulence scale. As general issue, DNS can be applied by solving the NS equations or some improved set of PDE.s that best fit the physics. And at the best I know, we are still not able to solve a full turbulence problem with characteristic scales ranging from the integral lenght to the Kolomogorov scale and extending up to shock thickness scale resolving the shock viscous structure (we can do it in analytical way in a localized 1D framework). That would be what I call DNS with shock-turbulence interaction.

I agree that a "better" continuum model could be used to better resolve shock structure, although at the moment such a model does not exist. As a physicist, I would be much more comfortable if microscale experimental data existed for the shock-turbulence interaction. However, I also believe that perhaps the details of the microscale dynamics of turbulence do not matter in the sense of the ideas of enslavement (e.g., Foias, Temam, Titi, et al.). Bohr once famously said that the job of physics is not to predict nature, but to predict what can be measured about nature.

Yes, I see and as an engineer my approach is not like a physicist...and the shock wave field (inviscid or viscous flows) is not exactly my field of expertise. At the best of my studies on the book of Landau (as well as other books) the NS equations are assumed to be a valid model for describing the shock thickness, provided that the shock is weak in such a way that its width is still sufficiently greater than the mean free path.

I would be glad if someone recently published that he was able to check the validity of the NS equations at least for shock at M=2.

Filippo: You might want to check out chapt 17 of that Landau Lifshitz book on fluid mechanics. In that chapter, they point out that NS is also missing the fluctuations. They propose modifications to NS that include stochastic terms for small scales. The resulting equations, now termed LLNS (Landau-Lifshitz Navier-Stokes) are used in the nanoscale community (lots of nice papers by John Bell and colleagues). It would be interesting to add those stochastic terms to a NS DNS and see what difference it makes.

Well, thanks Len for the reference, I have the second release of 1984, that unfortunately ended at Chap 16 ....

When you talk about fluctuations on small scales, you are considering those fluctuations as superimposition to the pointwise variable defined in the continuum hypothesis (that is, in more simple words, the superimposition to the DNS variable)?

Should not be them disregardable provided that the flow problem has the smallest characteristic lenght scale (i.e., the shock thickness) some order of magnitude greater than the mean free path? I can see that in nanofluidics you can encounter such issue but what would be the contribution for a problem at engineering scales?

If you add fluctuations to the original NS set of equations, the DNS approach seems to show a "closure problem" due to the non linear term, isn't that?

Thanks again for the discussion.

Filippo: I think the role of fluctuations is a big deal. For one thing, in the sense of dynamical systems, they lead to organization (Haken). There is a very nice paper by Chuck Leith in Phys. Fluids (1990) on Stochastic Backscatter illustrating this. For another, by the fluctuation theorem (Evans and Searles, 2002), fluctuations can generate regions of negative entropy production (over short times) that can flummox the regularized Euler schemes that you mention in a previous answer.

Am enjoying the discussion also. If you send me your email, I can return Chap 17.

Thanks, I just found the third edition and had a fast reading at Chap 17. If I am right, the main idea is about the modification of the Newton and Fourier constitutive laws (stress and heat flux relations) into the classical set of NS equations.

I can understand that approach as if the DNS NS equations were considered as governing equations for a resolved part and the modified Newton and Fourier fluxes act as unresolved and modelled terms in an LES-like manner. Obviously, considering the resolved and unresolved terms not in the sense of the LES filtering but in statistical sense.

But I am not aware of published papers testing such statistically-based DNS formulation for standard turbulence problems (homogeneous and isotropic turbulence, wall-bounded turbulence, etc.). Sorry but I cannot be of help about that proposal.

That is an interesting (and valid) way to think about it. Perhaps I could tempt you into an extension of those ideas. The Newton and Fourier constitutive laws come out of the collision integral in Boltzmann. Could similar statistical terms come out of fluctuations in the advection term? Of course the smallest scales are just what we call heat. But terms like the Reynolds stress result from fluctuating velocity fields that are not heat (in your LES analogy).

*

So consider a volume of fluid (could be a computational cell for example) and suppose that every "point of fluid" is governed by the Navier--Stokes equations. What equations govern the volume averages of the fluid (e.g., as would be needed for a finite volume numerical scheme)?

*

Those equations for the volume-averaged quantities cannot be NS, because of nonlinearity -- the product of the averages is not the average of the product.

*

I just put a paper up on RG that purports to answer that question. It may lead you to reconsider whether artificial viscosity is an escamotage. I would be interested in your thoughts.

Ok, thanks, I will read it. In line of what should be the governing equations, averaged over a volume, this topic has engaged me in the last years and I developed an integral-based formulation (weak-like approach) for expressing the congruent new representation of resolved and unresolved fluxes.

If you are interested in, please have a look at

Article What does Finite Volume-based implicit filtering really reso...

Article On the relevance of the type of contraction of the Germano i...

Let me some times to read and understand your paper.

Len, I read your article, actually I was already aware about MILES concept and the general present ILES approach. I also worked on that since 1992. As you know, turbulence community that follows the school of the CTR and some other researchers strongly advice the use of explicit closure physically-base model rather than an implicit closure.

To tell the truth, I have no problems in using both of them. I worked always using the integral-based formulation owing to the conservative property as well as on the suitable shape of the induced filtering that introduces a smoothing. Practically all my papers used to compare explicit and implicit closure.

I also can address the deconvolution approach we developed some years ago that is reported in a paragraph the Sagaut book.

Thus, I definitely understand your idea.

Filippo, Len: I read your entire exchange. Very interesting indeed, and I greatly appreciate its physical intent. Notwithstanding, to conclude the discussion directly pertinent to the original question posed by Shuang-Xi Guo, I'd say that regardless of the physical validity of NS equations, DNS can simulate unknown analytic solutions of the NS equations at any finite Reynolds number, and finite time of the solution evolution. Would you agree with this? Now, whether this is the "turbulence" measured in a lab or natural environments is a different story. Supposedly NS are modelled after the elasticity theory and have built in the linear Hook law into the strain rate formulation.

I totally agree!

Thanks a lot, Filippo!

Piotr and Filippo; I also agree. In the jargon of V&V, Piotr is describing 'verification', which is assuring that the computer is correctly solving the model equations. Convergence is an essential part of verification. 'Validation' is the complementary process of showing the converged results are physical, i.e., that they agree with nature. It is a separate effort, and can only be achieved by comparison with experimental data.

There are analytic solutions of full physics NS equations for shocks; Reisner and I have demonstrated that a (well-posed) numerical method can reproduce those results. However, those solutions and simulations do not correspond to experimental measurement. So for shocks, there is verification, but not validation.

The situation for compressible turbulence is murky because the required data does not yet exist. My answer to Guo's question is that DNS might reproduce physical turbulence, but that cannot be assumed. It must be demonstrated.

More than 100 years ago, Poincare wrote (in Science and Hypothesis) that there is no truth in mathematics, only consistency. He goes on to say the only truth lies in experiment.

Just my curiosity: when you say NS equations for shocks you actually mean to find a regular solution that has steep but continuous gradients across the thickness of the shock, isn't? And, using the DNS terminology, to resolve such gradienst we need the computational size to be at least an order of magnitude lesser than the shock thickness (while in classical DNS of incompressible turbulence we limit ourself to the Kolmogorov scale).

My question is if it was never performed such a DNS spanning the characteristic lenghts of the computational grid from order of a meter up to O(10^-8) m.

That would be the only way to assess if a DNS is validated in terms of experimental measurements.

But, just as a joke, often people want to validate the experimental measurements by using DNS ... :-)

Filippo: yes, a regular (strong) solution that has steep but continuous gradients across the thickness of the shock. In the paper "Entropy in self-similar shock profiles" with Reisner and Jordan, we re-derive this solution (with viscosity and heat conduction) which was first found by Becker. We ran the problem in 2D, generating the shock with a piston and putting about 25 cells across the shock profile and using centered differences with no artificial viscosity. There is no need to resolve a full meter of computational grid (which is a completely arbitrary length, why not a parsec?), only to use sufficient cells to allow the shock to reach its stationary solution. We also verified convergence.

Ok, I see and maybe I posed not correctly my question...Of course, one can simplify the problem solving directly the viscous Burgers model to get the viscous shock thickness or solving the motion of a piston causing coalescence into a shock. But I was thinking about a 3D case where turbulence is fully developed and energy spectra extends from a production term at an integral lenght scale (that I arbitrary set as 1 meter for unit of lenght) until to reach the shock thickness lenght. I don't have a relevant experience in viscous compressible flows and I would figure it out the shape of an energy spectra extending up to the characteristic wavenumbers of the shock.

There is no turbulence. Shocks are stable (self-healing) when running into a quiescent fluid. When a shock interacts with a density discontinuity (Richtmyer Meshkov instability for example), the contact discontinuity behind the shock may become turbulent but the shock front remains stable. Also, this is not viscous Burgers, which is much simpler than compressible Navier--Stokes.

Len I understand your thought. Turning back to your approach of using a modified set of NS equations for DNS, based on the integral equations, this is in line with the approaches I had in some of my previous papers on the deconvolution-based integral formulation.

You wrote in your paper that the governing equation in integral form is formulated for the locally averaged field and the equation is not closed due to the total flux that requires a reconstruction of the type F[u(u_bar)] (and its inverse). The integrated Taylor expansion can be approximately inverted and, for regular grid you get second derivatives. This has nothing to do with the local truncation error and artificial viscosity. However, due to the weak form, the second derivative in the flux is actually a third derivative in the strong form, that is a sort of additionally dispersive term. In other words, to have a viscosity-like term you would need the presence of a first order derivative in the flux. That would appear in the Taylor expansion only on non regular grid. Am I right?

Filippo:

1) No, sorry, but I disagree with your summary. The solutions for the volume averaged fields are strong solutions -- they are continuous and have continuous derivatives. I would categorize the concept of weak solutions as a mathematical abstraction with no physical relevance.

2) Nevertheless, I have been studying your paper and I do find many areas of agreement, most particularly in your discussion of the lack of commutativity of the averaging operator with the nonlinear fluxes.

3) Reflecting on Piotr's earlier comment, we have drifted far from the original question; our discussion is of great interest to me, but perhaps not to other readers. Should we find a different venue to continue?

Such a great and healthy discussion, it is definitely worth it to read each comment.

Len, Piotr, you are right sorry that somehow we shifted away the topic of the discussion form the original question. A new topic could be opened but the issues are so wide I am not sure neither about a suitable definition of the discussion. Something like "DNS in strong and weak forms"?

P.S. I did not write that the solution is piecewise constant due to the local averaging. On the contrary, it is continuous and differentiable in the sense defined by the convolution operation with the top-hat filter.

Len, keep me in the loop please. I could argue in defense of the weak solutions having a physical sense.

Independently, in 2012 Max Gunzburger had a very interesting seminar lecture in the Newton Institute on higher order extensions of the NS deformation tensor. These lectures are recorded. I'll find the link and send it via RG so other can also watch it.

Unfortunately, I have no idea how to save the relevant part of the discussion and copy in a new post on RG...

Piotr: by weak solutions, I mean those solutions to Euler's equations that shock up, forming a surface of discontinuity of density, velocity, etc. Such solutions do not exist as continuous functions (strong solutions), but belong to the space of distributions (weak solutions). I think these are mathematical idealizations, since a truly discontinuous solution cannot exist in nature. Perhaps you have a different meaning for weak solutions?

Len, I meant exactly what you mean; i.e., in the sense of the theory of distributions. I do not want to start new dispute, as it will take us to the problem of universals. I am attaching a very interesting paper by Salas relevant to this discussion.

Hi Piotrr: yes I am very familiar with Salas' review of the classical history of shocks, It shows how people fumbled around until the conservation of energy was properly understood. That review ends in 1910 when the papers by Rayleigh and by Taylor theory summarized the classical theory of shocks.. However, it was not until 1917 that the Chapman-Enskog theory predicted the connection of the macroscopic transport coefficients of viscosity and heat conduction to the microscopic theory of gas kinetics. And then Becker in 1922 who realized that the Chapman-Enskog theory predicted shocks would be less than one molecular mean free path wide. Little progress has been made toward a better continuum theory of shocks since then, though very good experimental data has been available since the 1960s.

* The work of Max Gunzburger on nonlocal calculus is closely related to Darryl Holm's alpha model and my own finite scale theory. I will look forward to the lectures you mention.

* Perhaps you mean that weak solutions are "useful" in the sense of George Box: "All models are wrong, but some are useful."

I am not sure if I understand you but my idea of weak form of the equation is based on a general mathematical projection of the equations along suitable differentiable test-function. We can recognize that the volume integral-based formulation of the conservative laws corresponds to the projection along a combination of Heaviside functions. In this sense we have a physical meaning of this specific weak formulation.

However, if we talk about NS equation we get into the fact that the diffusive fluxes still require differentiation property. Not sure how to generalize the definition of “weak.

Maybe you can open a post to better discuss...

Len, RE George Box: kind of, but paraphrased in the spirit of ~ all models are approximations, and some can be useful. Interestingly, Max refers to George Box in his talk that can be found on:

https://www.newton.ac.uk/event/amm/seminars , 9th October 2012 10:00 to 11:00. Note this is his second talk. There you can download slides or watch the recorded talk. I recommend the latter.

Dear Shuang-Xi Guo,

Being involved in experimental studies, I feel that the fundamental issue fall on the accurate representation of the initial and boundary conditions of the problem, provided numerical errors (round off and truncation errors) do not affect the results. Often even in experiments, the initial & boundary conditions are not accurately represented as we confine our results with some uncertainty. As you may aware that the fluctuations in velocity (disturbances) in subsonic flows (the disturbances propagate upstream through outlet) are caused due to many reasons such as roughness of the wall, vibration, free stream turbulence, curvature, buoyancy, local variations in ambient condition, etc. I do believe that DNS might provide accurate results due to its finest resolution if the numerical errors are ineffective (truncation after many digits) with correct initial and boundary conditions, and the governing equations are correct.

Any simple problem in supersonic regimes (mostly inlet conditions dominate the physic of the problem) can be chosen for comparing DNS and experiments as the external disturbances do not propagate into the field of interest.

thank you for all your interesting answers,

consult this dear Shuang-Xi Guo ,conslt this and good luck :

https://www.researchgate.net/post/Why_not_take_DNS_results_for_validating_turbulence_models_in_fluid_mechanics

Dear Tapan,

even if two years passed I don't think to be able to add more to that I wrote about DNS. I read something in the references you posted, perhaps from your writing I feel also you are quite skeptical about the present DNS results.

Does your feeling is more about the meaning of the numerical solution or about the real possibility to match all conditions of an experiment?

Or you are addressing the fact that a numerical solution can reflect a bifurcation from the non-linearity that an experiment can differently match?

I can highlight the doubt I read about the fact that small fluctuations can be relevant and some cases at moderate Reynolds number are no longer performed and validated using higher grid resolution to confirm the previous results.

I personally do not have computational power to make myself confident in performing high Reynolds numbers test-cases, I think that other people can answer and give you more insight than me.

I know what you mean. I had in past personal experience about groups using DNS code for solving complex problems (and publishing many papers) when such code simply failed any comparison on the plane channel flow, also in term of the zero-th order statistics!

But this history should deserve e different post about the "publish or perish" phylosophy ...

Turning back to your question, I never had the computational power to do such a DNS simulation about a transition problem over a flate plate, I hope some experienced researcher can share his experience.

I can add my idea that comparing the experiment of the transition along a wall in a wind tunnel is one of the most difficult task in DNS, owing to of the requirement of the correct BCs. In the experiment the plate is not semi-infinite and that causes perturbation from the end (outlet and lateral) of the wall as well as the inlet conditions should be matched.

I strongly believe this post is still alive, regardless the time elapsed. The last two inputs from professor Tapan and Denaro go straight to the point. Nevertheless, Professor Denaro highlighted something sad “publish or perish“. More recently I have seen this type of results with no novelty in top tier conferences. A large group of people are more interested on showing nice structures and color than physics and novel ideas. Back to the main point, I don’t think we are going to see this DNS computation soon because of the computational power required as Professor Denaro pointed out and also that “arena” remains as one of the most difficult and Finally, I dont see the founding agencies Interested on this topic. Therefore, researchers aim their weapons to publish what “will bring money in”.......Sad very sad.

I read the paper, I observed the particular numerical method adopted but I was not able to see the computational grid size. I also wonder what happens if F=0, does transition happen yet spontaneously just for the numerical errors?

My last question is about the possibility of LES to reproduce such physics. Assuming that a wall resolved BL is required, does a filtering in streamwise and spanwise directions would totally destroy such physics?

I would share my idea about the possibility to use LES in transition flow physics.

The main idea when LES was conceived is to filter out the small lenght scales posing the filter wavenumber in the inertial range of the energy spectrum. Therefore, we clearly address in this picture a fully developed turbulent state of the flow.

In transitional flow, we should start from the consideration that the turbulence is not fully developed, the energy spectrum is not correlated in the energy cascade and the filter wavenumeber is not constrained to lie in the inertial region. I think that the key is in the wavenumbers that are excited until to the transitional onset appears. Clearly if these wavenumbers are out of the description allowed by the filter wavenumber, their action is not resolved but acts only by means of the SGS model. Honestly, I don't know an SGS model that can describe such an excitation from the unresolved wavenumbers.

Thus, I think about a warning. Transitional flow can be simulated using LES provided that care is in the grid resolution. I think that a grid resolving the BL is mandatory. Filtering can be used in spanwise and streamwise directions provided that the excited wavenumbers are in the resolved range. We successfully used LES for transitional flows due to thermal BCs.

Of course if an SGS model would be able to model the amplification of the perturbation I would consider it.

some reasonable skepsis to dilute dns/les euphoria in this thread

https://www.researchgate.net/project/Third-order-multidimensional-Euler-method

Dear Anton,

I followed always with interest the enlighiting publications and lectures of Prof. Roe. However he is hystorically involved on solution of compressible Euler flow problems and we can never rigorously consider a DNS realizable for inviscid flows despite the fact that some theories for shock capturing method have several links to ILES

Dear Filippo,

I agree, that Roe deals with compressible inviscid flows. But removing compressibility or adding viscosity could not save. There are 2 gaps: between nature and Navier-Stokes and between analytical solution and numerical solution. I am not consider the first one. But the second one as a rule leads to "we need finer grids". Since 80-90th years there are calculations called by their authors "dns" however ending with demand of finer grid. And very rare thing is comparison of numerical solutions on several grids. Just "we used all resources that we have, here is the result".

As far as I know, there is no exact formula for Kolmogorov scale, just asymptotic estimate: constant* Re^(-3/4). That "constant" depends on everything except Re but is completely unknown way. So because we dont know the constant we dont know the Kolmogorov scale for a flow at hand. Therefore grid convergency is the only way to claim that the scale is resolved. I therefore wonder why simple thing grid convergency is reported so rare among dns papers.

By the way, about forementioned boundary conditions. I take part in many experiments, though hypersonic. In large amount of cases, turbulent flow downstream forgets about turbulization way. Was it due to surface roughness or freestream disturbances or someting else. So to model TRANSITION boundary conditions are important. To model TURBULENT flow they sre not important. There are exceptions of course. We will publish it later.

I agree that a grid convergence is a way to assess that a full resolution is achieved but you should also consider that some a-priori estimation of the the grid requirement is possible. Just as a rapid estimation, the cell Reynolds number is physically relevant to judge a DNS grid. If the Re_h is everywhere not more than one, you have that such computational lenght scale (numerically introduces the Nyquist cut-off) allows a perfect balance between the characteristic advection and diffusion time, that is exactly a way to define the Kolmogorov scale. At this level of resolution, also a first order discretization is able to produce a physically meaningful DNS. Refining further the grid would simply shift the Nyquist wavelenght below the Kolmogorov scale without changing the resolved physics. The numerical errors (diffusion and dispersion) have disregardable effects when they act only on the dissipative range.

As far as the BCs for different flow problems are concerned, this is a different issue that deserves a own post.

Yes, CFL has a strict connection to hyperbolic part of the PDE but in a practical computation, and that is also what you wrote above, the connection is in time and space and we should always consider the stability region CFL=f(Re_h). Apart from the numerical aspects of the stability, it implies also the range of dt and h sizes that allow all physical (advection, diffusion, etc) aspects to be well represented. According to the requirement Re_h=1 (a strong constraint), we should consider also the ratio u/(h/dt)

Dear Filippo and Tapan K. Sengupta

I do not want to disrupt anything. DNS, LES and RANS and quiet tunnels was worth things to do. But often done in a too optimistic way. So I just emphasize some forgotten by someone issues for whom who care.

I agree about your warning. However, while LES and RANS have some “Mystic or magician” assumptions in the modelling, DNS has some constraints to fulfill. If not satisfied one simply is not realizing a DNS even if uses this denotation.

Thanks, Filippo

Dear Tapan K. Sengupta

as you said, they are not telling the truth. Tell me me the truth they hidden please.

it is sad to my mind

https://www.researchgate.net/post/Could_quantum_mechanics_help_us_understand_turbulence

Counter question:

why don't we simulate Brownian motion and why Einstein presented the THEORY of BROWNIAN motion, and why Kolmogorov started a THEORY of turbulence? LES, DNS etc. are in selected cases a suitable solution.

And what do you do with DNS if molecular viscosity like in super fluids is zero?

Dear Helmut,

by definition, if the viscosity is zero you can never perform a DNS. A DNS is realized when you solve all local properties by taking the cell Reynolds number of O(1). Your simulation will be always a LES.

Yes Surely it will. DNS if performed with good accuracy and provided with accurate initial conditions along with suitable space and time discretisation can capture all range of length scale (largest eddy to the smallest eddy) thereby capturing the turbulent behaviour of the flow-field.

I'd like to say something as I understand: Surely you can obtain chaotic and stochastic results for DNS modeling of turbulent flow, as the solutions of DNS on N-S equations will produce chaotic solution. Please refer to the typical Lorentz equation solution for weather forecasting, in which the chaotic solutions were obtained. As a simplification of N-S equation, the Lorentz solution is a typical example for chaotic characteristic of modeling turbulent flow. And for stochastic, it is more easier to understand as you will obtain instantaneous solutions from DNS, and so you can do statistic analysis as you like.

However, I think the most challenging is how the turbulence occur with the chaotic and stochastic nature.

Any comments on my opinion are welcome. Thank you all.

Dear @Bin eventhough the solution field is stochastic and or chaotic the DNS solution is always deterministic. Whereas the topic mentions the fact that actual turbulent flows have different instantaneous solution field; thus non-deterministic.

Yes, DNS can solve turbulent flow problems, but it needs very big computer memory. If the Reynolds in order of 106, then we need memory in order of Re2.25 for each time step. So, the cell number is in the order of trillions. It is difficult to implement it in universities and industries.......

Dear researchers,

Have a nice weekend with your family....