I am coming back to the original question. Early turbulence depends, of course, on the ICs, but later the turbulence forgets its origin. For instance with spatial crossflow transition in a 3-d boundary layer, the crossflow vortices survive the onset of turbulence, but after some stretch they reduce to zero or a natural (unsteady) level that is given by the time-mean crossflow profile which still has an inflection point (similar to KH spanwise vortices in a turbulent shear layer).

In recent DNS of flat-plate flow it could be shown that the wake part of the turbulent u+ velocity mean profile is extremely sensitive to the Reynolds number where the inlet boundary condition is applied for spatially developing turbulence. This means that the influence is felt unexpectedly long. It turned out that for reliable DNS of spatially developing turbulence for any high Re number the inlet conditions (or the transition to turbulence) should be applied at a low Re number to get a unique result at relatively high Re number. Of course this rises the costs, but is necessary; see JFM 842, Wenzel & al. (2018) that I append.

Different "Bcs", for instance above the plate in the "free-stream", may alter the turbulence somewhat especially in supersonic flow. A wall above the plate (in an experiment) reflects emitted Mach and sound waves of the observed turbulent flow back onto it, possibly altering the turbulence structure depending on the actual situation.

Dear Tapan

interesting question, I am not able to give you an answer about published and non-confutable result.

I have my own opinion that turbulence problems admitting a statistically steady state could produce universal behaviours in the fully developed solution while that could be not true for flows in non energy equilibrium.

Your doubt is also somehow hystorical (Article Are Scalings of Turbulence Universal?

Waiting for further answers.

Dear Filippo Maria Denaro it is always a pleasure to hear from you. I know that the question is not resolved historically. However, reading many paper, it becomes apparent that many in the field assume (subconsciously!) that there is universality in the fully developed stage of turbulence. I will also patiently wait to hear other opinions.

What about your personal experience? I have the feeling form my experience that different excitations on the Poiseuille solution for canonical channel flows produce the same statistics in fully developed (and statistically steady) turbulence.

But when I tried some excitations for really unsteady fully developed flows (not in energy equilibrium) this was not so clear. But I did not focused anymore on DNS to get more experience and, as you know, using LES a lot of different problems arise.

I would be glad to know your opinion as well as those from researchers

I do not wish to prejudice others' opinion. For the same reason, we are looking at canonical zero pressure gradient flow excited (a) at the wall and (b) from the free stream. We will follow the transition and to nascent stages of turbulence. Apparently, the routes are different and the results are visually different. We do not have resources to perform DNS over a long domain to say anything definitively. We hope that someone out there listening and is motivated to do this. Are there any takers?

I am approaching this from an unbiased state and hoping to hear from others.

Dear @Tapan_K._Sengupta,

Here https://www.researchgate.net/publication/328721591_Regimes_of_flow_turbulization_near_swept_wing_edge_in_hypersonic_flow it is found that turbulent heat flux to the wall is independent on transition mechanism. Two types of initial disturbance are considered: artificial wall roughness and freestream nonuniformity. Hope it helps.

Dear Anton Noev,

A good data point for integrated quantity, such as heat transfer! Do you report statistics of fluctuating components? That also would be of interest.

Tapan, to sorry we did not measure fluctuations.

Probably, this lab's research would help https://www.researchgate.net/lab/Alexander-Kosinov-Lab

Dear Tapan, dear Anton,

just my two cents about the key of the question. If one supposes that the statistics of the fluctuating fields can diversify depending on the excitation route, does that mean one should observe (and measure) also different mean fields? Of course, I am talking always of statistically steady problem when the level of production and dissipation of kinetic energy balances each other.

I other words, while it is known that bifurcation of deterministic solutions can appear, does that is true also for the mean solution? It would be to admit that the steady RANS equations can get different solutions (for the same BCS.) even if they does not depend on an initial excitation...

Dear Filippo,

I am somewhat lost here! Can you please explain a little more, what you are suggesting? The main issue in my original question is: Are all fully developed turbulent flows are one and the same? If it so, then the route to turbulencce is not relevant and you can get to fully developed state by any correct/incorrect route, if you are thinking in terms of analytical and computational approaches! I am afraid that in computational approaches some groups indulge in lots of liberties in this way and in the end compare say skin friction or some such integrated properties.

I try to clarify better my observation. If we suppose that statistics are different, that implies (v'iv'j)_bar differs for each excitation route. Now, considering the exact RANS equation (I mean with the orginal Reynolds tensor, not wit a turbulence model), you can think that inserting the different statistics as exact known term you necessarily get a different mean field as solution.

In conclusion, for a statistically steady flow it seems we cannot admit different statistics of the fluctuations without admitting different resonse of the mean solution.

Am I wrong in this reasonment??

Assume we have some turbulent flow. To change it to some "another" turbulent flow we can introduce a perturbation into it - for example single surface roughness. For some length downstream, statistics and mean flow will be different from original flow if perturbation is large enough. But eventually far enough downstream turbulence will "chew" any perturbation and return to original statistics and mean quantities.

Another example is partial relaminarization: when turbiulent boundary layer comes through local high negative pressure gradient, it "partially relaminarizes" but eventually returns to original state. If interested, look for Narasimha's papers.

Anton: That reference does not prove anything! My question is about fully developed turbulent flow. It is not about relaminarization via perturbation.

Filippo: I understand that you are talking about the time-averaged governing equation. And you are reasoning that the "solution" must be unique. In many fluid flows, even for laminar flow, one notices multiple solutions and there is nothing wrong about it. Can we have more convincing logic and/ or proof?

yes, in deterministic sense we can have different laminar solutions but if we assume to study the fullly developed turbulence in term of the steady statistical solution (let me address not in term of the time averaging but in terms of N experiments, N->Inf), how can we admit that the exact RANS equations have different solutions (that means different statistics) for the same BCs?

Filippo: We are talking about different excitation, i.e. different BCs and IC. I would be perfectly comfortable with non-unique solutions!

since you wrote in the title of the topic the same geometry, I have assumed also the same bcs. and only different perturbed initial conditions...if you let the bcs be differently perturbed, provided that they produce different statistical bcs I totally agree about the possibility you are referring.

I am coming back to the original question. Early turbulence depends, of course, on the ICs, but later the turbulence forgets its origin. For instance with spatial crossflow transition in a 3-d boundary layer, the crossflow vortices survive the onset of turbulence, but after some stretch they reduce to zero or a natural (unsteady) level that is given by the time-mean crossflow profile which still has an inflection point (similar to KH spanwise vortices in a turbulent shear layer).

In recent DNS of flat-plate flow it could be shown that the wake part of the turbulent u+ velocity mean profile is extremely sensitive to the Reynolds number where the inlet boundary condition is applied for spatially developing turbulence. This means that the influence is felt unexpectedly long. It turned out that for reliable DNS of spatially developing turbulence for any high Re number the inlet conditions (or the transition to turbulence) should be applied at a low Re number to get a unique result at relatively high Re number. Of course this rises the costs, but is necessary; see JFM 842, Wenzel & al. (2018) that I append.

Different "Bcs", for instance above the plate in the "free-stream", may alter the turbulence somewhat especially in supersonic flow. A wall above the plate (in an experiment) reflects emitted Mach and sound waves of the observed turbulent flow back onto it, possibly altering the turbulence structure depending on the actual situation.

Thank you Markus! This is a very good demonstration of the point you are making. Wonderful piece of research and we congratulate you and hope to see more such results.

Regards, Tapan

Dear Markus, very interesting paper about the topic.

At this point I wonder also what about the role of the chosen flow model. The choice of the compressible or incompressible model could have some different impact? I mean, the compressible low-Mach model allows for a real pressure wave propagation describing properly the spatially evolving BL. The incompressible model just spread istanteneously the pressure perturbation for inlet through the BL. So, what about if we perform exactly the same geometry and BCs with the two different models?

Dear Filippo,

formulating proper boundary conditions for the incompressible-model DNS is indeed more subtle compared to compressible-model DNS because pressure disturbances are felt instantly everywhere by solution of some fully elliptic Poisson-type equations and related boundary conditions at any time step. Hence "same" BCs can only mean "similar in purpose". We were one of the first setting out a virtually non-reflecting outflow BC for incompressible DNS of spatial transition to turbulence, see ref. 1 below, where these hard demands could be fulfilled. (Slightly prior to that was Philippe Spalart with whom we discussed our differing models at that time). For DNS of transition to turbulence by forced perturbations at the wall we compared both models directly, see ref. 2 (page 6) below, and found no significant difference (as for physics), except that for the incompressible model the numerical background level was smaller in our case. Thereby the sound input ought to be minimized by choosing a wall perturbation that does not cause any mass input at any time step, and not, like a loudspeaker, only zero mass input over a perturbation cycle. (This is more than just "divergence free".) Also, sound waves are damped in the viscous region near the wall, and their propagation speed seems not to play a decisive role for the turbulence main physics. Hence if both models are carefully designed there are no significant differences for low Mach numbers. For turbulence e.g. induced by large roughness elements that show a feed-back loop , see for example ref. 3 below, there may be some difference for the "very exact" start of turbulence that however depends also on the numerical BC representation at solid walls.

Ref. 1: AIAA-J. 31 #4 (1993); Ref. 2: JFM 706 (2012); Ref. 3: JFM 796 (2016)

Thanks for the references.

Compared to the compressible flow model, I always considered the incompressible flow model much more complex in its mathematical formulation as well as in the implication of the underlined physics. Depending on the way we solve the coupled set of pressure-velocity field an intrinsic numerical perturbation is always induced at the wall. This is the case of the projection methods that, being based on the Hodge decomposition, introduce a systematic error in the tangential velocity component at the wall. Unfortunately, I have never seen studies that considered the influence of this error in the spatially developing turbulent BL. Generally it is only stated that the numerical pressure boundary layer does not interfer with the velocity field but that is formally true for an othogonal decomposition, which is not the case of the spatially developing BL.

Yes, the pressure formulation can be tricky if evaluated carefully. We used the 3-d vorticity-velocity formulation with 3 vorticity and velocity components for the full incompressible Navier-Stokes equations, and the pressure is not needed; it can be computed afterwards in case. The wall-vorticity calculation is the key point for a stable method, also with a proper outflow zone. The spanwise wall-vorticity component is computed in a downstream space-marching manner. The method is very reliable and fast except for strong suction or blowing at the wall: Then the wall-normal grid step dy must be substantially lowered to get the correct (converged) results, and the viscous time-step limit is dominating, enforcing a time-step going down quadratically with dy. With standard dy, as applied without strong suction (v_wall/u_inf

Yes, we also use velocity-vorticity formulation. In the process we have found spatio-temporal wave-front (STWF) to cause transition. TS waves comes into play for low frequency excitation only. The subject field is evolving and we continue to connect the dots...

Dear Markus, dear Tapan

I am not sure that using the vorticity-velocity formulation the counterpart of the tangential error disappears. Introducing the vorticity-stream function formulation I can say for sure not. It is just the other way the Hodge decomposition is solved. I have to think more about the vorticity-velocity method

Filippo Maria Denaro : I think that apart from the formulation, one may require to spend time in analysis of the numerical methods. Some of the evanescent structures will disappear due to "numerical stability" of the adopted method. For this reason, we have always advocated calibrating our methods with convection equation that demands perfect neutral stability. Without care spatio-temporal wave front (STWF) disappear. While capturing STWF is a plus point of a chosen method, retaining those for long time is also equally important.

Dear Tapan,

I agree about the requirement of the correct descripition of STWF, my doubts are about your initial question. If the excitation route by different BCs. drives to different statistics I wonder about the role of the chosen method. Actually, the vorticity-velocity (VV) formulation seems to me having the counterpart of the problem appearing in the pressure-velocity (PV) formulation. Disregarding form the present discussion the fully coupled methods, both VV and PV are based on the fact that the momentum equations writes in the form of the Hodge decomposition a +grad p = a* so that if you apply the curl operator apparently the pressure disappears. But, formally, when you integrate in time from t to t+dt you need to consider the time-evolution of the velocity field that requires again the pressure gradient. Explicit time integration apparently overcomes such an issue but in VV method while you prescribe correctly the velocity components on the boundaries in the elliptic equations for the velocity, the divergence-free constraint is not ensured. So you have to reintroduce this constraint in a different way, otherwise the error in div v term is implicitly an excitation in the kinetic energy (that we do not solve). My experience was (a lot of time ago), that such error is greater near the boundaries exactly as happens in the PV methods where the error in the tangential components is close to the boundaries and (eventually) cna spread into the interior. And that can be also analsyed in terms of a possible numerical stability issue.

It would be interesting to deeply analyse the presence of such errors and the possibile consequence in the description of the STWF.

Filippo Maria Denaro : The problem you allude to may not be present using stream function and vorticity formulation for 2D flow field. That is what we did originally in our 2011 PRL paper to show the existence of STWF. Both the velocity and vorticity are divergence-free (Solenoidal). Later on in 2014, we have shown STWF for 3D flow using VV formulation. Currently, we are also working with vorticity- vector potential formulation. Also note that for external flow, using continuity equation to satisfy divergence-free condition for velocity is perfectly safe and does not create any problem. This procedure does not work for internal flows- I completely agree.

Tapan K. Sengupta only apparently that issue does not appear but actually it does! the vorticity-stream function formulation is just the dual version of the Hodge decomposition. And that means you can only prescribe exactly one component. We are used to think that we can disregard such problem since by resetting the correct BC.s at each time step but we let the error spread in the interior in a sort of numerical boundary layer. As you wrote, that works perfectly only for inviscid flows.

DearFilippo,

Can you be more explicit? Apart from solving vorticity transport equation, one solves for the stream function equation, which is the kinematic definition of vorticity. What is meant by dual version of Hodge decomposition here?

Such a great discussion and topic. Dear professor Tapan, can you please provide us with the refernece you mentioned from 2011 ans 2014. Professoe Denaro, very interesting about about the double decomposition. If am I right, does it mean that Projection method comes from the same family of procedures as Vorticity -stream function formulation ? I thought that the lack of pressure term made the Vorticity -stream function extremly powerfull. But, it seems that it is not. I am a little lost !

The genesis of the error is quite sneaky, I spent a lot of time between 2001 and 2002 struggling to analyse the limits due to intrinsic errors of the PV and VV methods (you can see the Sec.3.3).

Article On the application of the Helmholtz–Hodge decomposition in p...

I debated two years with reviewers (they first rejected on a journal my analysis...) before getting my paper published in 2003. Now it is the most cited among my whole papers. I will try to give some ideas concerning the VV:

1) the time integration of the vorticity requires BCs in terms of the vorticity at the walls that are never exactly known but are approximated. As a consequence a first error is in the update of the vorticity field in the interior due to the BCs. Further approximations are introduced if one works with an implicit time integration.

2) The vorticity-stream function method is still based on the Hodge decomposition. The Poisson equation for the stream function is a consequence of a proper projector applied ont the acceleeration field, see Eq.(18). In this sense, the vorticity-stream function is the dual counterpart of the projection methods in PV formulation.

3) The solution of the Poisson equation requires only a condition on the psi function (extending to a vector 3D field). The source term is the updated vorticity, that is affected by the BC.s approximation and the solution of the elliptic equations does not ensure "simultaneously" that all the velocity components are satisfied on the boundary. We do not use the stream function to compute the velocities on a boundaries, but we disregard that result simply by setting the velocity we want to prescribe. But the solution of the stream function in the interior, close to a boundary, does not change and is affected by the error.

4) In the PV method, the counterpart is that we do not use the pressure gradient field to compute the tangential velocity on a boundary (summing to the intermediate velocity) but we set the natural velocity. But the pressure field has an error, appearing in the so called numerical boundary layer.

In conclusion, both stream-function/vorticity and velocity/vorticity are intrisically producing an error.

Julio Mendez PV and vorticity-stream function methods are from the same family, coming from the so called Helmoholtz-Hodge decomposition of a vector field in a bounded domain when its divergence, curl, normal or tangential components are prescribed.

Thanks professor Denaro I am going to read more about it, defintely ! Could uou please provide me with the reference ? You mentioned section 3.3. What reference should I refeer to?

Please bear with me, writting in this platform from my phone is a nightmare !!

Julio, the reference is linked above in my previous answer.

Filippo Maria Denaro:

I feel obliged to say that, upon repeated, extremely careful testing in the past, we could not find a palpable error in our finite-difference/spanwise Fourier-spectral vorticity-velocity method, for sure for simple geometry like the flat plate. Comparisons have been done for laminar instability in comparing the sensitive disturbance growth rates gained by the (unsteady of course) DNS with the ones from linear stability theory or the parabolized stability equations. Also direct comparisons with a compressible (conservative-variables )method have shown a collapse for small Mach number up to fully non-linear stages of transition and early turbulence. Naturally, the form of the Poisson equations for the velocities and the wall-vorticity BCs play a crucial role. The Poisson equations are formulated such that the continuity equation is in-built wherever possible, as well as the solenoidality of the vorticity vector; and only a small subset of possible combinations have been found to work properly. If blowing/suction at the wall is strong, then the wall grid step must be strongly reduced compared to a compressible method (due to the crucial wall-vorticity calculation). The extensive testing led then to the formulation we used (see the 2002 JFM paper on crossflow transition control).

Dear Markus,

how do you fulfill and verify the divergence-free constraint in discrete sense? The fact that with suction/blowing the grid step must be drastically reduced isn't a signal of the problem I am referring to?

Professor Denaro, I do not see the references in this post. Only professor Markus posted the references. Is that one of those ? Thanks Julio

Filippo: On the contrary, usage of three time level methods in your fractional step time step method is the source of error. We have now explained it in three papers (and also in my book) why no one should use three time level methods for governing equations with first time derivative. If you wish, then I can point out the references once again.. With three time level methods, you have two components of numerical amplification rates, one of which is the physical mode and the other is the numerical mode, which is of course spurious. I also completely in agreement with whatever Markus writes about VV formulation.

Tapan, yes, the type if time discretization introduces further sources of error but this is not the key of what I am addressing. The intermediate velocity field (or the vorticity field) provides only the vector field to be decomposed in the Hodge decompostion. We can alo address the discussion for a continuous exact vector field to decompose, no time integration at all. The core of the problem is in the nature of the mathematical continuous problem: you can set only one BC.

An example of the problem I use to clarify the issue is the 2d flow in a channel with a bluff-body solved with the stream-function vorticity. When you use the Hodge decomposition the problem is the elliptic equation with the vorticity source. Now you can set only one BC on the bluff-body. If you set the tangential component (no slip for the velocity) the Neumann BC will let the body be a possible source/sink, the normal velocity component is not set to zero. Conversely, setting a Dirichlet value allows for the body to be a stream-function with no traspiration. But the computed psi field close to the body cannot see the no-slip condition. Indeed, if you set arbitrary some constant value, you know it will be wrong when the body produces a vortex shedding at some Re number. The value of the psi over the body must cosnequently oscillate and its value is therefore an unknown of the problem that must depend on the interior field.

Formally, the same problem appears for the VV method. The Poisson equations for the 3 velocity components set the correct BC separately for each velocity component. Without introducing the discrete divergence-free constraint explicitly in the method you do not satisfy the continuity equation. This is what I asked before to @Markus, how they correct the method to fulfill the divergence-free constraint

Sorry Professor Filippo Maria Denaro the reference was in the message but I was not able to see it from my phone, I needed to use the web application (for the record). Thanks

Filippo Maria Denaro: As I said, please see the equations that we use, JFM 456 (2002), pages 53-57, or the NNFM paper (1999), see below. In the testing of the method we evaluated the continuity equation at each grid point after having computed all variables at each new time level to see the error as for divergence-freeness; the deviation from zero is very small with the equations and procedure used. In fact such, and checking the solenoidality of the vorticity vector, were the equations chosen, .

Thanks Markus I will read to understand better your method.

There is no universal modelsof turbulence. The special state of fluids when several forces are balanced and two or more solutions are possible to realize.

In future, am planning to give the world natural way of turbulence description.

Wait a little.

Will be extremely interested to read it! When do you plan to do it, Pavlo Lukianov?

For sub-harmonic and fundamental breakdowns, Sayadi and Moin (JFM, 2013) showed that both mechanisms relax toward the same statistical structure typical of developed turbulence. Although, for both scenarios, disturbances were introduced via continuous forcing at the wall.

Shirzad

Shirzad: Yes, they did similar what was done by the transition groups in Stuttgart (DE) and Tucson (US) 20 years earlier (Konzelmann, Rist, Fasel, and myself) by spatial DNS. What however was not done is turbulence at subcritical (with respect to modal instability) Re number, and in that regime 'turbulence' is very likely not unique.

Dr. Kloker: By subcritical Re number, do you mean Re number around or smaller than the Re number associated with the second branch in the modal instability (for given frequency)?

Before branch 1, non-modal or 'bypass' transition at low Re.

Agree with Markus! There are so many uncertainties about that paper and contents. I for one, do not trust IMEX method for DNS. It does not remain DNS with numerical noise. Also, the domain was so small, and moreover no TS wave was seen. That result should be reproduced by other competent group. You cannot do anything on Blasisus boundary layer and start calling it bypass transition.

Markus J. Kloker,

very interesting. Could you please provide links to papers on turbulence at subcritical Re. Either numerical or experimental.

Read this very recent paper on K-type and H-type transition in purest DNS form:

Three-dimensional transition of zero-pressure-gradient boundary layer by impulsively and nonimpulsively started harmonic wall excitation

• November 2018
• PHYSICAL REVIEW E 98(5)
• DOI:
• 10.1103/PhysRevE.98.053106
• 📷Pushpender Sharma
• 📷Tapan K. Sengupta
• 📷Swagata Bhaumik

Thanks, Tapan.

I briefly read the paper. It is interesting but not exactly what I asked for. First, the paper is about transition, not turbulence. Second, numerical dispersion has great impact on disturbances evolution so calculations should be performed on several different grids. Third, what Reynolds number is considered - sub or supercritical?

Anton: Since you seem to have cited a work that depends on two routes of transition to turbulence, you then must be aware that the lambda-vortex educed by Q- and \lambda-2 criteria show different vortical structures via these two routes. That should also tell you that your conclusion citing that work is utterly wrong! You got to be better informed and understand the basic differences that the PRE paper has vis-a-vis that JFM-2013 paper. I will not give you the ready-made answer. You got to read it better and not merely "briefly" and you will be very happy to find the answers yourself. That way one learns things better!! My advice to my students. But you are already an expert.

Dear Tapan,

What work did I cite?

I agree about carefull reading of papers. There is no data on turbulence in your PRE paper at all. If author cannot answer clear question, saying instead "look for it in my paper"... it is a bad sign. Scince should be clear to my mind.

Best regards

Interesting discussion but what about the original question of the topic?

First I assumed arroneously that the discussion focused on exactly the same flow parameters (I mean also consdering BC) and that only different initial perturbations should have been discussed. The fact that different perturbations inserted as BCs. can drive to different developed solutions is quite acceptable for me. If we assume a statistical solution based on the ensemble averaging (N->Inf) we should get different solutions as soon as we have different statistical BCs. for the problem. Otherwise the mathematical problem seems not well posed

Dear Filippo Maria Denaro

I also intuitively agree with you. Please refer to the preamble to the question. I am looking for some proof, theoretically or computationally. For example, if one performs the so-called frequency response experiment by wall excitation of a zero pressure gradient boundary at two different frequencies, which lead to two differently identified transition routes (known as K-type and H-type), then will the resultant turbulent flow will have same first and second order statistics (at least!)? Anton cited the paper by Sayadi et al (JFM,2013), which uses a tiny domain (does not even allow boundary layer to come out of the branch-II), uses an IMEX method (that creates upstream propagating disturbance at the interface of implicit and explicit domain inside the inertial layer) and fails to even get the well-known Tollmien-Schlichting wave, and yet the skin friction coefficient and shape factor matches with the experimental turbulence data given in Schlichting's book. While in another computation, these problems are carefully avoided and that shows a turbulent spot and that matches with the experimental data of skin friction and shape factor again. This is what perplexes me! You are saying that the results should be different, as the computational conditions are entirely different. Hoowever, the published accounts show the opposite. I hope that I have restated my question now. Any explanation?

Dear Tapan K. Sengupta

the JFM paper was cited by Shirzad Hosseinverdi, not by Anton Noev.

By theoretical point of view, if we assume the statistical solution to be representative of any possible realization of the same problem, the BCs for the statistical problem must be mathematically the same. Mathematics not Numerics is the real issue. The only way we can think of different statistical solution is by prescribing different statistical BCs, that is diferent flow problems. Now how we can suppose that the statistics of the fluctuations are different while producing the same averaged solution? That seems intriguing, is somehow like the problem of determining which is the primitive function when you provide only an average value: infinite solution.

I need time to think about...

Dear Anton Noev

My mistake! As spotted by Filippo that you did not cite any paper. I am perplexed by the situation mentioned in my previous post and the following answer. I hope you understand my condition. My apologies once again.

Thanks, Filippo!

Tapan, apology accepted, thanks to Filippo's clarification.:)

Anton Noev :

To your last question to me: see, e.g., Asai & Nishioka, JFM 297 (1995)

Boundary-layer transition triggered by hairpin eddies at subcritical Reynolds number, https://doi.org/10.1017/S0022112095003028, to cite an experimental paper.

Thanks, Markus.

has anyone first assessed the differences in the zero-th order statistics by performing a long time integration of the DNS solutions?

Dear FIlippo,

By zeroth order quantities, if you mean comparing skin friction coefficient and shape factor, then my previous post has given you an example. Both the published papers in JFM and PRE are supposed to be DNS (i.e. without any models) and both claimed excellent match with experimental results in Schlichting. Does this qualify for your test? I would be happy to see some independent tests. I hope that Markus is reading this!

Dear Tapan,

yes, something similar but not restricted to integrals at the walls. I was thinking also to compare the statistically steady velocity fields. Then, using these fields one can evaluate also the higher order statistics by the respective fluctuations fields.

Dear Filippo,

There is also a big difference in these two reported works. The JFM paper studied transition of Blasius profile, and thus did not have the leading edge of the semi-infinite plate in the computational domain. The PRE work obtained the equilibrium flow from the solution of Navier-Stokes equation in a domain that included the leading edge of the plate. Despite the transition to turbulence of different equilibrium flows, the integrated wall quantities turned out to be same! This has been bothering us for the last four years. I wish that some standard experimental results exists as benchmark, which is very accurately characterized to calibrate ANY claims of performing DNS, with FST measured in the tunnel, and first and second order statistics measured for the sole purpose of calibration. I think that this is also been discussed in RG in the context of other questions.