 Če so zelo, da ste v Romiji, priživaj, da se upraviti v vrših vrših. Sачu, da so prišli v, ok, na odličenji, na vziv-kod, na vziv-kod, o vziv-kod, Vse dobro vse ganjamo, da je bilo kompleksne, arbitraješke vse. Najbolj to je, da je to nečetne, nečetno je, da je to nečetne, Nače bomo seveda, da je veliko zelo, da je delaminarne vsega. Delaminarne vsega, kaj je drža, prvno vzelo na ranočke. A potem je... Oh, da je to več zelo. Da je to za smutne vsega. Zarenja izmeni za 所以a z všečenj Brinhti, vsak poslaré rehnulstvom vsebe. Proste, this is the other region of interest, that the data are mainly from the experiment, all the experiments of MikorAz, and then you will see that a certain point, the friction is constant, it doesn't depend anymore on the Rehnulstv number, ki je vsega odgleda na odgleda 8. OK, zame, da se vse začušajo, zato bomo vsega zelo, da je to vsega zelo, je zelo več sempljava. Našli da se prišli, da je renali se počuk, ali tudi nekaj neče zelo. Vsega zelo. In tudi, da je to vsega, neče zelo. češne jazvo, ko je učinitivno izlaminaj od laminarju, od turbovalent. Chčeš, da očinite, ki je očinitivno. Zelo je to, ki je ta, očinite z taj, učinitivno, to je naj očinitivno. Na svoj deli je taj, ki je učinitivno. So je to. In since we are using the DNS, the direct numerical simulation, this region is the most appropriate region to study. because we have a very good tool. Instead, also, experimentally, it's quite difficult to measure accurately what is going on in transitional flows. OK, zato je to, da smo vse izgledali. Vse zelo se, da smo nomerikali, in nekaj nekaj, zelo se, da je zelo, zelo, v uvaj, da je zelo, p je stendiril na pjah, chah je, dish retro diz wedding chelo ABs, asis de square chelo S. Uč nek primer zumetje volim betrayed both v square chelo and both for pipe, and also for the channels known to for three. Per umetje v svej držav,üler중etショja iman grass ki prim aiming of the turbo and pipe agree quite well with all the data that were obtained by a very high Reynolds number numerically. So, it is what I am saying. So I am interested to this region, that is there is a zoom here, and then here you would see that we have a lot of simulation, we did a lot of simulation with square čanel, ki je tukaj tukaj v tem vseh lečju. In tukaj vseh vidimo, da je tukaj čanel s ovenim symbolem, da je tukaj čanel s solitim symbolem. Zdaj je tukaj dobro. Tukaj je kartizija korda, tukaj je pola korda, zato vse, da srečimo, srečimo, ovo smo vseh tukaj. Ok, zato vseh tukaj vseh vseh tradizijačna zelo vizono vseh vseh, da bi tukaj, da bila izvukovana uroda z turboljstvom energijstvom. Turboljstvom energijstvu kineki turboljstvom energijstvu je tukaj vseh dobro tukaj vseh tukaj toga tukaj. O, ok, Sorry, maybe I skip one. Ok, this is the turbulent energy. This is the turbulent energy for the channel at this one. And then you will see that when there is a transition, there is a sharp growth of the turbulent kinetic energy. Then the turbulent kinetic energy is decreasing, increasing the Reynolds number. And then you would see that there are two power law that we didn't try to understand why there are two power law, but then you would see that there are two very distinct power law. And then if we are going to the pipe, that this is the pipe, also you would see that there is a transition, quite sharp transition, and then there is a power law here. And usually the Reynolds number of the simulation for the pipe is smaller than one for the channel because, you know, in polar coordinates these simulations are more computer time demanding. And for the square channel even more. And then now, let's say instead, when you have a square channel, here you would see that the transition is quite smooth, is different from the other two flows. And then here you will see the red and the green are one for the Cartesian and one for the polar coordinates of the cylindrical coordinate. And then you will see that there is a good agreement that this could be a check that doesn't matter which coordinate system you get, the same results. OK, now let's say turbulence, kinetic turbulence is composed of three components, u prime, v prime, w prime. I am in favor of the v prime. The v prime is a quantity that was never too much analyzed, mainly because the experiment is impossible to measure or very difficult to measure the velocity. And then, but you have to remember this stress, normal stress is the only one entering in the momentum equation. Only in the v, u, streamwise velocity, is entering the Reynolds stress, u prime, v prime. But this is the only one. And why is interesting? Because we know turbulence is interesting because there are structure. If we are able, which is the quantity now that is accounting for the structure, so that should be important. OK, and then you will see which are the structure. We know that there are ribon-like structure where there is sij, sij, and then there are vertical, let's say, rod-type structure that are more given by omega squared. We know that in homogeneous turbulence these two are equal, and so there is no anything interesting. All bounded flows instead of these two are important. So you will see from this equation that the secondary derivative of the normal stress is the one accounting for the structure. And then you will see here, and then you will see here, here is a channel, smooth channel and this is a rough channel in order to see how they change the structure. Here you can see that increasing the Reynolds number, you will see that we are going to have this structure that are increasing the strength of these ribon-like structure that are increasing. But then to me, you are saying that transition is interesting. Look, when you are going to very high Reynolds number, this is Reynolds tau, 4,000 and then 2,000, you will see there are not too many changes. Here, look how many changes, how is changing the structure near the wall. And then they are going, we are going to have a turbulent when they are going to survive. And then also here you will see that here there is a transition because between these ribon-like and these round, then this is the region. When there is an interaction between these two kind of structure, you have the maximum turbulence energy production that we know that is around y plus 12. OK, here the other things that you will see that in near rough wall you will see there is the structure, you destroy the structure. The structure, we know, we did a lot of simulation and we showed that near the rough wall you are going to increase the isotropy, you are decreasing the anisotropy and so everything should be more simple. And this is what maybe was observed by Nicoraz. And since I am always advocating in the importance of V, I was able for this rough wall to derive theoretically, let's say, a moody diagram that is related to the V. So when if you are increasing the V, the V at the plane of the crest at any distance from the wall you are going to increase the friction and then you will see here we cannot go to the flat region because V is not this equivalent 8 of the walls introduced by Nicoraz. Ok, this is to tell you our importance. Now again, we are now to look and understand why there is this, we are going from laminar to turbulent. Here we have the radial velocity for a circular pipe. Here you will see that going from Reynolds 1500 to Reynolds 2000 you are going to increase the V and then you see that this structure, this is related to the structure, these structures are very big so they cannot move and so the turbulence cannot grow. I think in order to grow you have to have the freedom. Without freedom you are killed and this is valid everywhere in politics everywhere. I think then you will see that at a certain point you have the freedom and then you start to have this structure and there are two kind of structures there are the near wall structure and the other big structure that they are always connected one to the other and then you know here I don't show but then I think this structure in near the wall or any wall the turbulent in wall in near wall bounded flows are related to the ratio sq over epsilon are two times k. Only when this is large you have the growth of this structure they can survive or to be present and then here you will see that in order to have this big structure sq over epsilon should be greater than 5 and then near the wall this is still higher and then they are stronger this structure and so this is the one important because they are going to give you the friction. Now if we are going now to look at the other channel now there is the square channel and then the square channel you will see the transition is different because at 1500 still you have this one and then you will see that is going to change from 1500 to 1000 and then you will see that this is something related to the secondary motion near the corner of the square channel and then you will see that at a certain point this again is what I am trying to say that it is not large changing with Reynolds number, you will see the only thing that you can appreciate is the peak is moving close to the corner and that is another important thing and then let's move to do, I told you that turbulent energy is important and then you will see that the difference between circular pipe and square channel, this is only one fourth you understood that it was doing the average also with the other four corners 1500 there is no turbulent energy 2000 there is no turbulent energy and then at 3500 in me there is a turbulent energy that is distributed and then you will see that in this case I am averaging only in time and in x in this downstream direction and then you will see that there are still peak around here so that means this secondary motion what you want to call they are permanent they don't go everywhere because if they were moving like that then they should be uniform instead they are coherent structure because they are coherent if you are to be coherent you don't have to change in time for a long period of time then for example maybe if you are running the simulation for years then they are going to be more you lose these coherence but then in in our in our in fluid mechanics we have to be coherent for a long period respect to the simulation to the time scale related to the boundary ok and then you will see how it is different when we start to have turbulent energy and then you will see what is going to have before it is everywhere and then it is going to be always becoming closer to the wall so you will see also here but the center and so you will see the region here, the total summing everything is bigger than the other the other two Reynolds number ok now sorry again and ok, this is the effect of the Reynolds number for the pipe and for the Reynolds also in this case you will see that a very high Reynolds number everything is moving always closer to the wall but then I think not too much happening here now let's let's go to the channel ok the comparison between the square channel and the two dimensional channel this is the u plus u plus in the channel all the simulation I did going from laminar to turbulent completely you will see that there is this is the laminar y square and then you have always at the end you have the log low if there is a long log low this is another point but then you will see that in the square channel now I am measuring the velocity up to the besetrics of the accounting in the two direction and so you will see that depending of the Reynolds number I have all the I am going from the I with the log low to region very similar to those and then you will see the same is happening here the same is happening here so the square channel maybe will be a very interesting tool to understand the wall turbulence because there are the interaction between the very region very high very high Reynolds Reynolds style and very low Reynolds style and so that I think maybe we can understand more this the wall bounded flows with that what I am doing here is I am scaling the quantity with the local friction the friction is varying along the wall is zero at the wall at the corner and then is increasing going towards the center ok and then you will see and then you see the same Reynolds dependence in a square channel let's say this is a square channel at Reynolds 20000 and then you will see a very high Reynolds number depended similar to the one that we have for the large number of simulation I was doing ok and then let's say the same is for the stream wise vorticity the same Reynolds number so this is in order to tell you that maybe the square channel is a very interesting tool to use to understand the wall turbulence ok, now looking at the stress we want to see the stress and then this is you will see also here there are two stresses in the momentum equation the momentum equation and then you will see here that these T12 and T13 are similar they sum but then I am not able from this picture to understand we know that in wall bounded flows in the channel in the pipe we have the total stress that is linear this is the reason why I did also the simulation comparing the simulation in Cartesian coordinate the one is in polar coordinate you will see that they are quite similar there is still something going on here I am not converging it fully converging so this is the now ok, this is the total stress and then you will see now that I am able to have the total stress that I call TRZ the other one that is the TZT that is going should sum to zero and then you will see that there is how they behave one is the viscous stress near the wall the other one is concentrated everywhere and then so this is the Reynolds number dependence and then let's say this is the total stress still you will see that here we have the Reynolds 6000 and then this 10,000 at 30,000 and then you will see that there is a tendency to the straight and the line was obtained by the pressure gradient and then that I need in order to have steady state solution and then you will see that is going to that the difference should be related maybe should be related to the maybe I didn't run long enough and then but then you will see that the other stress this TZT is almost going to zero it should go to zero but then again we have also we have to do some interpolation to evaluate this one no interpolation but then some around take out some point where there is a solid that is in the circular should is coming out then we know the streaks this is what we are interested these are the streaks for the circular pipe these are the streaks for the square channel and then you will see that here which is you will see again there are these elongated structure this is omega r and the other one is omega r then you will see there are some thin layer of this vortizi near the wall do it to curvature but then again when it is becoming turbulence there are these elongated structure that we are going to see a very high Reynolds number there is also the reason why I said why Reynolds number is important we have all these doesn't change too much instead when you have transitional region you start to have these that are changing a lot and then ok this is to conclude because I finish and then ok so we are going now to do more that changing the aspect ratio these are picture of stream wise velocity and then also we are going to analyze this also this kind of flow pipe that are corrugated pipe and then you will see only the last one are the so called the riblets and the riblets you know they could be drag reduction we have the tool to understand the better wall turbulence any wall turbulence that's it I think it's a working procedure yeah but then why there are also turbulent flow I know but this is also for that one it's valid for everything if you are going this is the reason because I'm interested to see how is the interaction between these that you call mean flow I know that there is UV but then in the other case but so everywhere also in any any only for the smooth wall we know that in the mean you can average ok but here also also here I can average I can always average the hydraulic the ammeter is in the Reynolds bulk but then I am not completely that is important the hydraulic the ammeter and then this is the result so if you look at that there are simulations with the same area so if I am I am thinking more that there is a paper of somebody that is saying the important is the area so you have to build a length based on the area more than on the hydraulic the ammeter because the evosplut this is the reason why we are doing this so still it is open it is not completely clear because also experiment I don't think there are many experiment with complex all these kind but then you understand but the important things we are doing the simulation keeping constant the area because in the flow in the bulk you have the area you understand but then you have to see but this is a numerical experiment if you believe in the numerical experiments not only because we are using staggered grid with the staggered grid if this is the grid the normal velocity the other velocity is there the other one is at the center of the cell so there is no singularity I think on the equation we have only boundary condition you understand we don't have to solve the equation at the corner but then that is difficult because you know I am doing this the simulation in polar coordinate to take account the complex surface I am doing the merce boundary technique but then if it is bounded I don't know because my this pitch before me she was doing something like that I don't know how they were doing that because if you are using general could be linear coordinate the numerics is tricky I spend a lot part of my life it was very difficult especially for in the incompressibility condition it is a tricky that is the reason why I move to polar coordinate or other things it is a tricky or other things so that will be more deep I don't want to go there sorry