Hi everyone. I would just like to thank everyone who noticed the typo in Equation 30. I wanted to confirm that the term in the brackets should be -2/3 dU_k / dx_ k and not -1/3 dU_k / dx_k. This error has been carried forward into equations 32, 34 and 36 but is correct in equations 39 and 40. I have left the video up in its original form and have pinned this comment, so that you are all aware of the typo. For my patrons on Patreon, the lecture slides have been corrected and you can find the correct version of the lecture slides there to download. Thanks again everyone for spotting the typo!
For high Reynolds number flows, resolving the fluctuations in time is too expensive, even with a big computer. A common approach is to time average the NS equations and model the effect of the high frequency oscillations, rather than resolve them. Time averaging is really only necessary because computers aren't fast enough to resolve turbulence at high Reynolds numbers 🙂
I don't think -1/3 dU_k / dx_k is incorrect. Infact there should a term 1/2 outside mu_t x {dU_i / dx_j + dU_j / dx_i -2/3 dU_k / dx_k}. This is because you missed a 1/2 on the RHS of equation 9. This changes eqs 39 and 40 as well. Probably that is the reason why OF doesn't have 2 in front of (nut_)*dev.
Have you ever wanted to clap and cheer in the middle of a movie or a concert, thrilled by the quality of what you are experiencing? As a teacher, this was my feeling as this lecture was going on. Wow, I wish I could explain things as this lad! Aidan, you are definitively gifted. Thank you very much for generating and sharing this invaluable material.
Continue to be impressed by your clear and well spoken lectures on everything around CFD. A fantastic resource & setting a standard. This collection will no doubt add fantastic high end quality to the currently available study aids and theory across the topic of CFD.
This video and his explanations are as beautiful as the physical phenomenon and math modelling he is trying to explain! This is a gift to humanity! :) Please keeping more of these videos Dr. Aidan.
I also went down the turbulence modelling rabbit hole for my Master's. Thank you for the effort to convert your research into useful and understandable slides. This is a considerable amount of work!
Another 5-star lecture, thank you so much for sharing your knowledge! I now see that the eddy viscosity model is a key enabler beneath the turbulence models we so often rely upon; you filled in an important missing component in my understanding of RANS formulation.
You have given me so much more confidence in my work. I have been working in electronics packaging design for aircraft, and often need CFD to understand and define system performance. I have a mechanical engineering background, but not much in fluids. With each video, a new corner of Fluent is demystified. You have my thanks and respect for making these excellent videos which present complex concepts in a highly digestible formats.
Hi Dr. Aidan, may I add some remarks for the Boussinesq approximation here: 1. Reynolds stress (RS) term is sub-divided into isotropic and anisotropic components. 2. For the isotropic component of RS term: the axial components of the RS term are summed up and related to turbulent kinetic energy. The assumption is that: this part of the turbulence is assumed to be isotropic!! It somehow makes sense since it is on the isotropic component of RS. 3. For the anisotropic component of RS term (which is subtracting the full RS term by its isotropic component), this component is analogue to the diffusion term of the N-S equations while the eddy viscosity is introduced to replace the dynamic viscosity. In most of the RANS turbulence models (except the RS model), the eddy viscosity is the same in all the axial and shear components of this anisotropic component of RS term. Again, isotropic turbulence is assumed. But this time - isotropic turbulence assumption is on the anisotropic component of RS!! This is a plausible reason why RANS model is not that accurate in some situations where turbulence is very anisotropic!!
great work man, I have been working on my engineering project in CFD and after a week of exploration on the internet found your videos on various CFD models and their basics. it's helping me a lot in better comprehension of basics. thanks, man!
In the end of this video you stated what you wanted to achieve with this video, and let me assure you that you did accomplish just that. Thanks for the incredible content!
Great job. I had studied this stuff last semester during my PhD. That 1/3 thing is tricky because depending on the source I have seen, they use the same notation for S and S* (only S for example), then we ask ourselves "where the hell this 1/3 comes from?". Again, great job. It was really nice to rediscover this and refresh my memory.
Finally after 3 years of modelling CFD, I finally understand! Thanks Dr. Aiden. However I must admit that I did struggle in understanding the first few slides. I had to refer to Dr. Steve Brunton's YT derivation to prepare me to understand your initial slides
Thank you Dr. for this excellent lecture. Explanation was pretty lucid and comforting despite lot of mathematics involved. This lecture has certainly helped in building a strong foundation for further learning the turbulence modeling. Thank you again 😃
Thank you so much. I have been in a hole trying to learn this from contradicting resources with differing notation or incomplete explanations (I'd never heard of the Kronecker Delta and was scared of this random symbol appearing with no explanation). I was in deep despair for my masters thesis but i think i have some hope now. Star
You did a wonderful job deriving the eddy viscosity formula. Thanks very much! You made it very easy to understand. I should say you nailed it! PS. I noticed that other people mentioned it and you pinned it as well but would like to emphasize that the missing "2" on the left side of equation 9 or "1/2" on the right side, affects the proceeding equations. If you consider OpenFOAM formulation, that is the reason there is no "2" behind (nut)*dev ...
Typo is sort of a curse, never goes away, always follow us, lol. As I said earlier, you have done a great job explaining behind-the-scenes of CFD codes. Good luck!
Thanks, the lecture perfectly reveals the idea of turbulence modeling :) But I've a little misunderstanding. At the 19th minute, we equate the symmetric components of the Reynolds stress tensor and obtain formula (9). But if they are equal, then when they are added, we should get 1/2*mu_t(dU/dy+dV/dx). I would be grateful if someone could explain this issue.
Hi Aidan, you said at about 12:50 that momentum is transferred in the direction of the velocity gradient but it should point upwards and not down so I think is more correct to say that momentum is transported against the velocity gradient
Correction @12:53: Since dU/dy is positive, the direction of velocity gradient is upward. So should the statement not be "momentum is transported in the direction of negative velocity gradient"?
Hi Dr. Aidan!. Thank you once again for the wonderful lecture. At 12:49, isn't the gradient supposed to point towards the maximum? Please correct me if I am wrong.
Yes you are correct. Sorry if I wasn't clear! Momentum is transported down to the lower particles, which have lower momentum. This is transport from high to low (towards the minimum). This is why there is the minus sign in the front of the Reynolds stress, because the momentum is transported in the direction of negative gradient. Pesky minus signs always catch me out 😅
Thanks for the fantastic lecture. 13:03 isn't the momentum transported in the opposite direction of gradient? this is confusing to me. If considering the direction of gradient, the direction in which the function increases most quickly from, then the momentum like other transport quantities like mass and heat points toward the opposite of gradient of a quantity (velocity, temperature and mass respectively).
Ok, the pinned comment is correct as OpenFOAM has a documentation page on Linear eddy viscosity models that shows that as well, but if that is the case then something will definitely be wrong in the early eqns of 8 or 9 I guess.
Very impressive, Sir! I have now understood very well on eddy viscosity modelling and its derivation to obtain a correct value to solve the momentum equation for the velocity field from your presentation, fantastic effort with complete clarity in your presentation!...keep doing this Sir, you are a blessing to many who venture into CFD. Would your be able to do one on Coupling of Level-set method and VOF model for two-phase flow interface tracking technique?
At 20:12 , by the logic of Eqs (7) and (8), shouldn't Eq.(9) have a factor of 2 (or half)? This would then translate into the uu components described in the next slid (eq. 10 onwards). Perhaps something else needs to be factored in rather than just symmetricity?
You have to kiss a lot of frog videos to have your prince video. You have just made my life easy, this is the best video lecturer on this topic that I have watched. I am saying it after watching somewhere around 15 other videos.
At 27 minutes, isn't there an error there in the notation? Either it should be: -rho*u'_i*u'_i on the LHS, or a factor of 2 need to appear on the RHS. What am I missing here?
Hi Dr Aidan I have a question in equation number 9 should the eddy viscosity multiply by 0.5? see the video in 20.00 Because we add equation numbers 8 and 7 to get equation 9. Am I right?
Great video, as always! Indeed, finding consistency notation in CFD Turbulence modeling is challenging , especially at the beginning of the study. Associated with "magical jumps" from one equation to other (not only in papers), it was really a problem. Until now, for my dissertation I've been using the notation present in the book "Turbulence Modeling for CFD" from D. C. Wilcox.
Hi Dr. Wimhurst, I would like to point out a few things: 1) First off, there's a typo in equation 10. The RHS has to be multiplied by a factor of 1/2. That means equation 12 should be without that factor of 2. 😃 2) Secondly, I fail to see what's "inconsistent" in equations 15 & 16 too. If in equation 15, you "should get" -2ρk on the RHS, and in equation 16, you "are getting" the μ_t (∂U/∂x + ∂V/∂y + ∂W/∂z), doesn't that simply mean you've found an alternative form/derived expression for k (after equating the RHS in both equations)? So, basically I don't understand why we have to force it like this by adding & subtracting the 1/3rd terms? Please correct me if I'm wrong. 3) I don't understand how you came to the following conclusion that: a) that there's an "error" when we add the normal stress components (I mean that's just simply algebraic addition of 3 equations right, so what errors are we talking about?).... hence leading to an overprediction of the μ_t (∂U/∂x + ∂V/∂y + ∂W/∂z)..... hence leading to the underprediction of k. Greatly appreciate your helping the CFD student community 😊😊
Hello Aidan, Thanks for the amazing video again, my compliments! In Equation (9) though, I think a 2 factor is missing at the left hand side. Am I wrong? Thanks. Mauro
Wow... you always come up with some great videos... and rightly said it will help me immensely in my master thesis which i am doing now 😛 Great admirer of your work 👏
Thanks for the video. It did clear up a few conceptual issues, but I'm still left with one. You have shown that to calculate how U, V, W change with time, it is not necessary to know , and
I have a question - it is more of an OpenFOAM one but since you mentioned OpenFOAM here, I am tempted to ask it. Apologies if it is not relevant. In the momentum equation of say simpleFOAM for example, i notice this term turbulence->divDevReff(fvc::grad(U)) which is the deviatoric part of the stress tensor without the 2/3 TKE. Now when I use a turbulence model such as SA model where k is not explicitly calculated, I need to use a postProcessing utility (such os simpleFoam -postProcess -func R). But i get the message that TKE is not defined for SA and hence I get the Reynolds stress tensor WITHOUT the subtraction of the TKE (I know this because if I take the trace of the obtained Reynolds stress tensor, it comes out to be zero, same as the start of the video which resulted in the mathematical manipulation). I suppose I have two questions - firstly, is this Reynolds stress tensor even correct? Secondly, is it true that to get TKE, one must run a two equation turbulence model such as k epsilon or k omega SST and cant obtain it from a post Processing step? I got these questions only after watching your video.
Great explanation, thank you very much Aidan! I'm just wondering if around minute 12 there is a little error about the direction of the gradient of U that you mention (downwards). I understand that the gradient is rather pointing upwards and the transfer of momentum, which is counter-gradient, is pointing downwards. Looking fotward to your feedback!
The net momentum transport is in the opposite direction of the velocity gradient right? Cuz momentum transfer is from top to bottom while velocity gradient points from bottom (lower) to top (high)
Great video and very clear explanations. However, I can't get past eqns 7 and 9. How can both equations be simultaneously true unless one of the velocity gradients is equal to zero. To come up with eqn 9, I thought you would add eqns 7 and 8 together. If that is true, then there would be a factor of 1/2 on the RHS of eqn. 9. What am I missing??
I would love to see a video on why the limitations pop up and what we can do about that. I have a model where all three of the cases you warned about show up all over the model (bending tubes, baffles, conical reducers etc.). Are there any models that can handle that? K-omega SST seems to give physical results, but how accurate can the data be?
The best you can really do is to compare to experimental measurements and see which gives the closest agreement. It is hard to say which will be most accurate until you have some results that you know are true (for comparison). Personally, k omega SST always seems to be a solid choice for me and it's what I normally pick if I am unsure
Hi, thank you for the wonderful video. I am confused in equation 21 why is the term 1/3(dU/dx + dV/dy +dW/dz) necessary? To me it seems that the continuity condition would make this term zero. Thank you!
This term will only be zero for incompressible flow 👍 most authors retain this term so that the Boussinesq model is suitable for compressible flow as well. If you are using incompressible flow, you are only adding zero, so there is nothing to worry about 👍
Does this apply to compressible flow aswell? Shouldn't the Navier-Stokes equation (1) contain the extra piece you get from constitution equation for shear stress for gases?
You rock! Thanks It strikes me that the Reynolds normal stress "correction" is a bit ad-hoc and not exactly physically motivated. Oh well, seems to work!
Hey Aidan. I am little confused on the normal stress derivation. I see you agree that equation 9 should have 1/2 on the right side. And you derive the normal equation by swap v with u in equations 9. Is that means then equation 11 should have 1/2 on the right side too? That means the coefficient 2 will be canceled in equation 12. Furthermore, equation 16 and 17 should not have coefficient 2 on the right side too. That leads to equation 21, 22 and 23 should not have 2 in front of nu_t. This eventually leads to the point that equation 30 should have 1/3dU_k / dx_k. This is conflict with your pinned message. where did I derived incorrect? I feel I am missing something. Could you please let me know? Thanks a lot!
Do you have a video with an explanation of the physical meaning / relative importance of the different terms of the Reynolds stresses ? Like, how is the magnitude of the pressure term -2/3 rho. k compared to 2.mut.dU/dz, and also their respective signs ? Thanks a lot, great lecture.
I think in 23:30 you intentionally add 1/3( continuity eq) in each eq 21-23, however, there was no need since cont equation = 0 so 1/3 contribution doesn't make any sense
A big Thanks for this Masterclass. However I have a question about equation 39. We have 2 in factor, then we get 2/3 of ∇ . U instead of 1/3. Isn't it a pb ? Maybe I will find my answer after writing my comment but i take the risk
![[CFD] The Smagorinsky Turbulence Model (Part 1)](http://i.ytimg.com/vi/V8ydRrdCzl0/mqdefault.jpg)
![[CFD] The Smagorinsky Turbulence Model (Part 1)](/img/tr.png)
![[CFD] The k - epsilon Turbulence Model](http://i.ytimg.com/vi/fOB91zQ7HJU/mqdefault.jpg)
Hi everyone. I would just like to thank everyone who noticed the typo in Equation 30. I wanted to confirm that the term in the brackets should be -2/3 dU_k / dx_ k and not -1/3 dU_k / dx_k. This error has been carried forward into equations 32, 34 and 36 but is correct in equations 39 and 40. I have left the video up in its original form and have pinned this comment, so that you are all aware of the typo. For my patrons on Patreon, the lecture slides have been corrected and you can find the correct version of the lecture slides there to download. Thanks again everyone for spotting the typo!
Please, make a video on Variational multiscale turbulence modelling
Why is it necessary to do time averaging to original NS equation??
For high Reynolds number flows, resolving the fluctuations in time is too expensive, even with a big computer. A common approach is to time average the NS equations and model the effect of the high frequency oscillations, rather than resolve them. Time averaging is really only necessary because computers aren't fast enough to resolve turbulence at high Reynolds numbers 🙂
@@fluidmechanics101 That's so cool
I don't think -1/3 dU_k / dx_k is incorrect. Infact there should a term 1/2 outside mu_t x {dU_i / dx_j + dU_j / dx_i -2/3 dU_k / dx_k}. This is because you missed a 1/2 on the RHS of equation 9. This changes eqs 39 and 40 as well. Probably that is the reason why OF doesn't have 2 in front of (nut_)*dev.
Have you ever wanted to clap and cheer in the middle of a movie or a concert, thrilled by the quality of what you are experiencing? As a teacher, this was my feeling as this lecture was going on. Wow, I wish I could explain things as this lad! Aidan, you are definitively gifted. Thank you very much for generating and sharing this invaluable material.
Thank you so much, I really appreciate it
Continue to be impressed by your clear and well spoken lectures on everything around CFD. A fantastic resource & setting a standard. This collection will no doubt add fantastic high end quality to the currently available study aids and theory across the topic of CFD.
This video and his explanations are as beautiful as the physical phenomenon and math modelling he is trying to explain! This is a gift to humanity! :)
Please keeping more of these videos Dr. Aidan.
Thank you so much for your kind words, I really appreciate them
You solved some of my longstanding problems with turbulence with this absolutely fantastic video. I can't thank you enough.
I also went down the turbulence modelling rabbit hole for my Master's. Thank you for the effort to convert your research into useful and understandable slides. This is a considerable amount of work!
This is by far the best video on turbulence. Can’t thank you enough!
Far and wide the best explanation i have seen.
Another 5-star lecture, thank you so much for sharing your knowledge! I now see that the eddy viscosity model is a key enabler beneath the turbulence models we so often rely upon; you filled in an important missing component in my understanding of RANS formulation.
You save my life!!! As a beginner in CFD simulation, I am so confused with equations. Your lecture do enlighten me. Thanks so much
You have given me so much more confidence in my work. I have been working in electronics packaging design for aircraft, and often need CFD to understand and define system performance. I have a mechanical engineering background, but not much in fluids. With each video, a new corner of Fluent is demystified. You have my thanks and respect for making these excellent videos which present complex concepts in a highly digestible formats.
Hi Dr. Aidan, may I add some remarks for the Boussinesq approximation here:
1. Reynolds stress (RS) term is sub-divided into isotropic and anisotropic components.
2. For the isotropic component of RS term: the axial components of the RS term are summed up and related to turbulent kinetic energy. The assumption is that: this part of the turbulence is assumed to be isotropic!! It somehow makes sense since it is on the isotropic component of RS.
3. For the anisotropic component of RS term (which is subtracting the full RS term by its isotropic component), this component is analogue to the diffusion term of the N-S equations while the eddy viscosity is introduced to replace the dynamic viscosity. In most of the RANS turbulence models (except the RS model), the eddy viscosity is the same in all the axial and shear components of this anisotropic component of RS term. Again, isotropic turbulence is assumed. But this time - isotropic turbulence assumption is on the anisotropic component of RS!!
This is a plausible reason why RANS model is not that accurate in some situations where turbulence is very anisotropic!!
Great points! Thanks for your help 🙂
I don't need any books anymore :) Everything is clear after your videos. I love your style and diagrams!
great work man, I have been working on my engineering project in CFD and after a week of exploration on the internet found your videos on various CFD models and their basics. it's helping me a lot in better comprehension of basics.
thanks, man!
What an amazing lecture... Thank you so much for preparing this material and for sharing it with us.
In the end of this video you stated what you wanted to achieve with this video, and let me assure you that you did accomplish just that.
Thanks for the incredible content!
Awesome lecture Dr. Aidan ... Thanks a lot for the time and effort to make this amazing content available for free here in youtube!
That was unbelievable. I understand it perfectly now. Your Lectures are greatly appreciated.
thank you for filling all the empty slots in my brain with these beautiful derivations :D helped me massively!
the best intro of eddy viscosity model online, thanks
Great lecture. Very helpful towards understanding basics of CFD modeling
FM101 is *Digital Gold* for CFD Community
Great job. I had studied this stuff last semester during my PhD. That 1/3 thing is tricky because depending on the source I have seen, they use the same notation for S and S* (only S for example), then we ask ourselves "where the hell this 1/3 comes from?". Again, great job. It was really nice to rediscover this and refresh my memory.
Amazingly simple and to the point explanations!
Simply immpecable. Believe me I haven't have enough words to praise your efforts.
Thanks for your clear explanation for the derivation of the eddy viscosity model
Finally after 3 years of modelling CFD, I finally understand! Thanks Dr. Aiden. However I must admit that I did struggle in understanding the first few slides. I had to refer to Dr. Steve Brunton's YT derivation to prepare me to understand your initial slides
Thank you Dr. for this excellent lecture. Explanation was pretty lucid and comforting despite lot of mathematics involved. This lecture has certainly helped in building a strong foundation for further learning the turbulence modeling. Thank you again 😃
concise and quality content. you are one of the rarest🙌. thanks for the tutorial.
This was the best explanation what is the basis of the 2 eqn model. Simply amazing, precise and concise
Great Lecture, I always love your practical explanations and insights into the theoretical models
Thanks Dr Aidan for these wonderful insights
I'm really enjoying the video series, thank you posting this Dr Aidan!
Thank you so much. I have been in a hole trying to learn this from contradicting resources with differing notation or incomplete explanations (I'd never heard of the Kronecker Delta and was scared of this random symbol appearing with no explanation). I was in deep despair for my masters thesis but i think i have some hope now. Star
You did a wonderful job deriving the eddy viscosity formula. Thanks very much! You made it very easy to understand. I should say you nailed it! PS. I noticed that other people mentioned it and you pinned it as well but would like to emphasize that the missing "2" on the left side of equation 9 or "1/2" on the right side, affects the proceeding equations. If you consider OpenFOAM formulation, that is the reason there is no "2" behind (nut)*dev ...
Such an annoying typo for me to make 😅
Typo is sort of a curse, never goes away, always follow us, lol. As I said earlier, you have done a great job explaining behind-the-scenes of CFD codes. Good luck!
@@fluidmechanics101 Well if that is true then it changes some equations going ahead.
One amazing lecture! Thank you so much, Dr. Aidan!
I'm a graduate student in CFD and machine learning -- thank you for this!!
Perfecto! Your les series chapter is outstanding! Keep up the good work!
Really Helpful ..and how you addressed it from very basic to advanced. It is really really an informative presentation. Thank you very much
This is an amazing job, well-done Doc.
Thanks a lot. I did a bit research recently on this but wasn't successful. Thank you for lay out this so clearly.
Excellent job in sorting this all out!
Thanks, the lecture perfectly reveals the idea of turbulence modeling :)
But I've a little misunderstanding. At the 19th minute, we equate the symmetric components of the Reynolds stress tensor and obtain formula (9). But if they are equal, then when they are added, we should get 1/2*mu_t(dU/dy+dV/dx). I would be grateful if someone could explain this issue.
I think the same and thus going forward certain equations will have a missing factor of 2.
Hi Aidan, you said at about 12:50 that momentum is transferred in the direction of the velocity gradient but it should point upwards and not down so I think is more correct to say that momentum is transported against the velocity gradient
The clearest physical explanation and mathematical derivation you'll find anywhere on eddy viscosity models applied to RANS modeling
Correction @12:53: Since dU/dy is positive, the direction of velocity gradient is upward. So should the statement not be "momentum is transported in the direction of negative velocity gradient"?
Hmmm yes that does make sense 🤔
brilliant, Thank you from KSA, Riyadh.
Thank you for high quality video!
I'm also waiting for your Reynolds stress model video :)
Hi Dr. Aidan!. Thank you once again for the wonderful lecture. At 12:49, isn't the gradient supposed to point towards the maximum? Please correct me if I am wrong.
Yes you are correct. Sorry if I wasn't clear! Momentum is transported down to the lower particles, which have lower momentum. This is transport from high to low (towards the minimum). This is why there is the minus sign in the front of the Reynolds stress, because the momentum is transported in the direction of negative gradient.
Pesky minus signs always catch me out 😅
@@fluidmechanics101 Hahaha... I would like to request you to continue the series of LES. Thank you!
Bravo thanks what a great lecture you’re really an amazing lecturer teacher, commentor UA-camr Thank you.
Thank you for your kind words, I really appreciate it
Amazing work, your videos have been helping me so much lately.
Best explanation, it captures everything needed to understand eddy viscosity
Thanks for the fantastic lecture.
13:03 isn't the momentum transported in the opposite direction of gradient? this is confusing to me. If considering the direction of gradient, the direction in which the function increases most quickly from, then the momentum like other transport quantities like mass and heat points toward the opposite of gradient of a quantity (velocity, temperature and mass respectively).
Clear and concise. Excellent.
Keep on that good work ! Many thanks from a Fluid Mechanics lover
Yes, this is exactly what I was looking for! :D
Brilliantly clear explanations. Thank you!
can someone please explain to me how we got equation 9, in my understanding we should divide the term on the right by 2
Ok, the pinned comment is correct as OpenFOAM has a documentation page on Linear eddy viscosity models that shows that as well, but if that is the case then something will definitely be wrong in the early eqns of 8 or 9 I guess.
Masive help for my aero class! Will probably buy your course
Very impressive, Sir! I have now understood very well on eddy viscosity modelling and its derivation to obtain a correct value to solve the momentum equation for the velocity field from your presentation, fantastic effort with complete clarity in your presentation!...keep doing this Sir, you are a blessing to many who venture into CFD.
Would your be able to do one on Coupling of Level-set method and VOF model for two-phase flow interface tracking technique?
At 20:12 , by the logic of Eqs (7) and (8), shouldn't Eq.(9) have a factor of 2 (or half)? This would then translate into the uu components described in the next slid (eq. 10 onwards). Perhaps something else needs to be factored in rather than just symmetricity?
I think this is part of the error which I have noted in the pinned comment. Well spotted! There is indeed a factor of 2 missing
Really great content and so clear explanations. Bravo !! And thanks !
You have to kiss a lot of frog videos to have your prince video. You have just made my life easy, this is the best video lecturer on this topic that I have watched. I am saying it after watching somewhere around 15 other videos.
Thank you very much for your kind words 🙂
Thank you very much for your hard work and effort.
At 27 minutes, isn't there an error there in the notation? Either it should be: -rho*u'_i*u'_i on the LHS, or a factor of 2 need to appear on the RHS. What am I missing here?
I had the same thought
Ahhh yes! The factor of 2 has gone missing somewhere along the way 😩let me look into this and correct it
youtube needs more content like this. very useful
Hi Dr Aidan
I have a question
in equation number 9
should the eddy viscosity multiply by 0.5? see the video in 20.00
Because we add equation numbers 8 and 7 to get equation 9.
Am I right?
Great video, as always!
Indeed, finding consistency notation in CFD Turbulence modeling is challenging , especially at the beginning of the study. Associated with "magical jumps" from one equation to other (not only in papers), it was really a problem. Until now, for my dissertation I've been using the notation present in the book "Turbulence Modeling for CFD" from D. C. Wilcox.
It's a great book! I think sticking with that notation is a good approach. Good luck with your dissertation
@@fluidmechanics101 Thank you! :D
Great explanation! Very clear as usual
Hi Dr. Wimhurst, I would like to point out a few things:
1) First off, there's a typo in equation 10. The RHS has to be multiplied by a factor of 1/2. That means equation 12 should be without that factor of 2. 😃
2) Secondly, I fail to see what's "inconsistent" in equations 15 & 16 too. If in equation 15, you "should get" -2ρk on the RHS, and in equation 16, you "are getting" the μ_t (∂U/∂x + ∂V/∂y + ∂W/∂z), doesn't that simply mean you've found an alternative form/derived expression for k (after equating the RHS in both equations)?
So, basically I don't understand why we have to force it like this by adding & subtracting the 1/3rd terms?
Please correct me if I'm wrong.
3) I don't understand how you came to the following conclusion that:
a) that there's an "error" when we add the normal stress components (I mean that's just simply algebraic addition of 3 equations right, so what errors are we talking about?).... hence leading to an overprediction of the μ_t (∂U/∂x + ∂V/∂y + ∂W/∂z)..... hence leading to the underprediction of k.
Greatly appreciate your helping the CFD student community 😊😊
Hello Aidan,
Thanks for the amazing video again, my compliments!
In Equation (9) though, I think a 2 factor is missing at the left hand side. Am I wrong?
Thanks.
Mauro
Yep, you are right. Some other people spotted this as well. Just a typo!
@@fluidmechanics101 Dr. Aidan, wouldn't it affect the rest of the equations? I think that that missing factor affects equation 30 and subsequent ones.
I think my pinned comment should explain this 👍
@@fluidmechanics101 Thnks for reply Aidan.
Great Explanations!!! Thanks mate
Wow... you always come up with some great videos... and rightly said it will help me immensely in my master thesis which i am doing now 😛
Great admirer of your work 👏
Parabéns! I am always looking forward to see your videos.
Thanks for the video. It did clear up a few conceptual issues, but I'm still left with one. You have shown that to calculate how U, V, W change with time, it is not necessary to know , and
Cmu = 0.09 ( an empirical constant). K and epsilon are the calculated by solving 2 transport equations. This is the basis of the k epsilon model!
@@fluidmechanics101 Thanks. Is there another video I can reference for how to do that?
Yep, just check out my video on 'The K Epsilon model'
Big thanks for video! That's awesome.
Really great session.
Thanks.
Will you have another video for the non-linear eddy viscosity models?
I have quite a long list of videos to make before then, but if I get some time then yes 🙂
I love your work and your videos! Keep this incredible work 🙏
Thanks for the video! @20.19 if eq. 9 is derived by adding eq. 7 and eq. 8, then shouldn't we have 2 in the denominator on RHS?
Yes. A typo.
I have a question - it is more of an OpenFOAM one but since you mentioned OpenFOAM here, I am tempted to ask it. Apologies if it is not relevant. In the momentum equation of say simpleFOAM for example, i notice this term turbulence->divDevReff(fvc::grad(U)) which is the deviatoric part of the stress tensor without the 2/3 TKE. Now when I use a turbulence model such as SA model where k is not explicitly calculated, I need to use a postProcessing utility (such os simpleFoam -postProcess -func R). But i get the message that TKE is not defined for SA and hence I get the Reynolds stress tensor WITHOUT the subtraction of the TKE (I know this because if I take the trace of the obtained Reynolds stress tensor, it comes out to be zero, same as the start of the video which resulted in the mathematical manipulation). I suppose I have two questions - firstly, is this Reynolds stress tensor even correct? Secondly, is it true that to get TKE, one must run a two equation turbulence model such as k epsilon or k omega SST and cant obtain it from a post Processing step? I got these questions only after watching your video.
Very good and useful as always 👌
Great explanation, thank you very much Aidan! I'm just wondering if around minute 12 there is a little error about the direction of the gradient of U that you mention (downwards). I understand that the gradient is rather pointing upwards and the transfer of momentum, which is counter-gradient, is pointing downwards. Looking fotward to your feedback!
You could be right. It has been a while since I put together this talk and I remember this bit being really confusing!
Amazing explanation!
The net momentum transport is in the opposite direction of the velocity gradient right? Cuz momentum transfer is from top to bottom while velocity gradient points from bottom (lower) to top (high)
Great video and very clear explanations. However, I can't get past eqns 7 and 9. How can both equations be simultaneously true unless one of the velocity gradients is equal to zero. To come up with eqn 9, I thought you would add eqns 7 and 8 together. If that is true, then there would be a factor of 1/2 on the RHS of eqn. 9. What am I missing??
I would love to see a video on why the limitations pop up and what we can do about that. I have a model where all three of the cases you warned about show up all over the model (bending tubes, baffles, conical reducers etc.). Are there any models that can handle that? K-omega SST seems to give physical results, but how accurate can the data be?
The best you can really do is to compare to experimental measurements and see which gives the closest agreement. It is hard to say which will be most accurate until you have some results that you know are true (for comparison). Personally, k omega SST always seems to be a solid choice for me and it's what I normally pick if I am unsure
Hi, thank you for the wonderful video. I am confused in equation 21 why is the term 1/3(dU/dx + dV/dy +dW/dz) necessary? To me it seems that the continuity condition would make this term zero. Thank you!
This term will only be zero for incompressible flow 👍 most authors retain this term so that the Boussinesq model is suitable for compressible flow as well. If you are using incompressible flow, you are only adding zero, so there is nothing to worry about 👍
@@fluidmechanics101 thank you!
Does this apply to compressible flow aswell? Shouldn't the Navier-Stokes equation (1) contain the extra piece you get from constitution equation for shear stress for gases?
Great video !!!! But a small doubt isn't momentum transported opposite to the direction of velocity gradient?
Thank you very much. It was amazing 👏👏👏👏
You rock! Thanks
It strikes me that the Reynolds normal stress "correction" is a bit ad-hoc and not exactly physically motivated. Oh well, seems to work!
Exactly
And in eq.30 why are the indices on the RHS different for the first two terms? earlier we had du/dx and dv/dy, i.e. the indices were the same.
Hey Aidan. I am little confused on the normal stress derivation. I see you agree that equation 9 should have 1/2 on the right side. And you derive the normal equation by swap v with u in equations 9. Is that means then equation 11 should have 1/2 on the right side too? That means the coefficient 2 will be canceled in equation 12. Furthermore, equation 16 and 17 should not have coefficient 2 on the right side too. That leads to equation 21, 22 and 23 should not have 2 in front of nu_t. This eventually leads to the point that equation 30 should have 1/3dU_k / dx_k. This is conflict with your pinned message. where did I derived incorrect? I feel I am missing something. Could you please let me know? Thanks a lot!
I agree with Hulala, @Fluid Mechanics 101 could you explain?
Check my reply on pinned comment. You are right.
Do you have a video with an explanation of the physical meaning / relative importance of the different terms of the Reynolds stresses ? Like, how is the magnitude of the pressure term -2/3 rho. k compared to 2.mut.dU/dz, and also their respective signs ? Thanks a lot, great lecture.
I think in 23:30 you intentionally add 1/3( continuity eq) in each eq 21-23, however, there was no need since cont equation = 0 so 1/3 contribution doesn't make any sense
Really great! Thanks for everything!
A big Thanks for this Masterclass. However I have a question about equation 39. We have 2 in factor, then we get 2/3 of ∇ . U instead of 1/3. Isn't it a pb ? Maybe I will find my answer after writing my comment but i take the risk
Yes, I think you might be right 😅
@@fluidmechanics101 Ha ! so it's 1/6 ?
Thanks for this. Could you do a video on turbulence models that are not eddy viscosity models, like cubic k-epsilon?