[CFD] Eddy Viscosity Models for RANS and LES

[@fluidmechanics101]

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!

[@anurajmaurya7256]

Please, make a video on Variational multiscale turbulence modelling

[@anupamdas8277]

Why is it necessary to do time averaging to original NS equation??

[@fluidmechanics101]

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 🙂

[@leslienovihoho4658]

@@fluidmechanics101 That's so cool

[@parasghumare8067]

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.

[@ana.scarabino]

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.

[@fluidmechanics101]

Thank you so much, I really appreciate it

[@scugglebottom]

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.

[@sangeeth49]

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.

[@fluidmechanics101]

Thank you so much for your kind words, I really appreciate them

[@gerhardbekker646]

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!

[@brandongleeson9903]

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.

[@AliMostafazadeAbolmaali]

You solved some of my longstanding problems with turbulence with this absolutely fantastic video. I can't thank you enough.

[@arthurwittmann6242]

Far and wide the best explanation i have seen.

[@productivity_station]

This is by far the best video on turbulence. Can’t thank you enough!

[@sudhanshumaurya2860]

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!

[@CalvinoBear]

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.

[@leminhduy8368]

You save my life!!! As a beginner in CFD simulation, I am so confused with equations. Your lecture do enlighten me. Thanks so much

[@engineeringarts4509]

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!!

[@fluidmechanics101]

Great points! Thanks for your help 🙂

[@lenacaban4178]

I don't need any books anymore :) Everything is clear after your videos. I love your style and diagrams!

[@brunaveras734]

What an amazing lecture... Thank you so much for preparing this material and for sharing it with us.

[@14johnnymarques]

Awesome lecture Dr. Aidan ... Thanks a lot for the time and effort to make this amazing content available for free here in youtube!

[@cronos864]

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!

[@sangal666]

That was unbelievable. I understand it perfectly now. Your Lectures are greatly appreciated.

[@atakan716]

thank you for filling all the empty slots in my brain with these beautiful derivations :D helped me massively!

[@sienliu663]

the best intro of eddy viscosity model online, thanks

[@EngLhag]

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.

[@bijendersingh43]

This was the best explanation what is the basis of the 2 eqn model. Simply amazing, precise and concise

[@idreeskhan-zp5ey]

Amazingly simple and to the point explanations!

[@mustafamarvat863]

Simply immpecable. Believe me I haven't have enough words to praise your efforts.

[@wonsungjin]

Thanks for your clear explanation for the derivation of the eddy viscosity model

[@jessbuildstech]

I'm really enjoying the video series, thank you posting this Dr Aidan!

[@realkanavdhawan]

FM101 is *Digital Gold* for CFD Community

[@leslienovihoho4658]

Great Lecture, I always love your practical explanations and insights into the theoretical models

[@AmitMishraUP43]

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 😃

[@prestonr6348]

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

[@vivekkhantwal8346]

concise and quality content. you are one of the rarest🙌. thanks for the tutorial.

[@reza1815]

Perfecto! Your les series chapter is outstanding! Keep up the good work!

[@roubert1979]

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 ...

[@fluidmechanics101]

Such an annoying typo for me to make 😅

[@roubert1979]

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!

[@parasghumare8067]

@@fluidmechanics101 Well if that is true then it changes some equations going ahead.

[@andrewgibson7797]

I'm a graduate student in CFD and machine learning -- thank you for this!!

[@davidwang8270]

Thanks a lot. I did a bit research recently on this but wasn't successful. Thank you for lay out this so clearly.

[@ajinkyapawar6638]

Great lecture. Very helpful towards understanding basics of CFD modeling

[@vsjhade]

Really Helpful ..and how you addressed it from very basic to advanced. It is really really an informative presentation. Thank you very much

[@solaadekoya9836]

This is an amazing job, well-done Doc.

[@pritampriyadarshi4530]

Thanks Dr Aidan for these wonderful insights

[@j.gordonleishman6401]

Excellent job in sorting this all out!

[@francescoindolfo]

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

[@yazanatrash]

Best explanation, it captures everything needed to understand eddy viscosity

[@narenbharatwaj7854]

One amazing lecture! Thank you so much, Dr. Aidan!

[@stophercry]

Amazing work, your videos have been helping me so much lately.

[@jimdeligiannakis6314]

Clear and concise. Excellent.

[@jack-buckhilll5428]

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?

[@prakashthirunavukkarasu29]

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.

[@fluidmechanics101]

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 😅

[@prakashthirunavukkarasu29]

@@fluidmechanics101 Hahaha... I would like to request you to continue the series of LES. Thank you!

[@colonelManyBears]

Brilliantly clear explanations. Thank you!

[@노랑몽스터]

Thank you for high quality video! I'm also waiting for your Reynolds stress model video :)

[@CarlosEduardo-xr8eu]

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.

[@fluidmechanics101]

It's a great book! I think sticking with that notation is a good approach. Good luck with your dissertation

[@CarlosEduardo-xr8eu]

@@fluidmechanics101 Thank you! :D

[@Thescavenger4]

Keep on that good work ! Many thanks from a Fluid Mechanics lover

[@Lbarkway]

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

[@meshalnafea5200]

brilliant, Thank you from KSA, Riyadh.

[@dodo19961375320]

youtube needs more content like this. very useful

[@francescomadonia4922]

Great explanation! Very clear as usual

[@ssss4391]

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"?

[@fluidmechanics101]

Hmmm yes that does make sense 🤔

[@sauravpanda1380]

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 👏

[@sitrakaforler8696]

Really great content and so clear explanations. Bravo !! And thanks !

[@slim590]

Thank you very much for your hard work and effort.

[@JAYasankarPillai7]

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.

[@fluidmechanics101]

Thank you very much for your kind words 🙂

[@gabrielpinananaharro7154]

Masive help for my aero class! Will probably buy your course

[@DiegoAndrade]

Bravo thanks what a great lecture you’re really an amazing lecturer teacher, commentor KZbinr Thank you.

[@fluidmechanics101]

Thank you for your kind words, I really appreciate it

[@michaelmello42]

The clearest physical explanation and mathematical derivation you'll find anywhere on eddy viscosity models applied to RANS modeling

[@Darkkenfox]

I love your work and your videos! Keep this incredible work 🙏

[@MalcolmAkner]

Yes, this is exactly what I was looking for! :D

[@VishalSingh-os5oj]

Great Explanations!!! Thanks mate

[@nikhildarekar672]

Really great session. Thanks.

[@maurotortora5317]

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

[@fluidmechanics101]

Yep, you are right. Some other people spotted this as well. Just a typo!

[@Luiferhoyos]

@@fluidmechanics101 Dr. Aidan, wouldn't it affect the rest of the equations? I think that that missing factor affects equation 30 and subsequent ones.

[@fluidmechanics101]

I think my pinned comment should explain this 👍

[@klandestineabnun]

@@fluidmechanics101 Thnks for reply Aidan.

[@turalsuleymanov4529]

Amazing explanation!

[@БабичЕлена-б6ш]

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.

[@parasghumare8067]

I think the same and thus going forward certain equations will have a missing factor of 2.

[@abzrg]

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).

[@goodday276]

Very good and useful as always 👌

[@hishamlotfy3942]

Will you have another video for the non-linear eddy viscosity models?

[@fluidmechanics101]

I have quite a long list of videos to make before then, but if I get some time then yes 🙂

[@ilyassebouatou3161]

can someone please explain to me how we got equation 9, in my understanding we should divide the term on the right by 2

[@jhongoulart9493]

Parabéns! I am always looking forward to see your videos.

[@troyanych1]

Big thanks for video! That's awesome.

[@hamedheydari989]

Thank you very much. It was amazing 👏👏👏👏

[@abzrg]

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.

[@rishikeshmishra9485]

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?

[@fluidmechanics101]

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

[@orlandopalone80]

Perfect, as always

[@MalcolmAkner]

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?

[@yazanatrash]

I had the same thought

[@fluidmechanics101]

Ahhh yes! The factor of 2 has gone missing somewhere along the way 😩let me look into this and correct it

[@asifjaved9797]

Really appreciate your work Sir. Could you please make a video on Navier stokes equation? This would help those who are beginner in the field of CFD. Looking forward for the video.

[@MalcolmAkner]

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?

[@fluidmechanics101]

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

[@alexz26389]

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

[@fluidmechanics101]

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!

[@alexz26389]

@@fluidmechanics101 Thanks. Is there another video I can reference for how to do that?

[@fluidmechanics101]

Yep, just check out my video on 'The K Epsilon model'

[@raffaellavuolo8389]

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!

[@fluidmechanics101]

You could be right. It has been a while since I put together this talk and I remember this bit being really confusing!

[@shobob29]

Thank you for a wonderful lecture!!!

[@ferasalgafary9063]

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?

[@MG-hm9bh]

Really great! Thanks for everything!

[@HolzmannCFD]

Thank you for your effort preparing these nice talks. Refreshing my mind and is in correlation with my book :)

[@nwsteg2610]

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!

[@fluidmechanics101]

Exactly

[@diwasbajracharya3745]

New guru in town.

[@hulala6097]

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!

[@elchupaka]

I agree with Hulala, @Fluid Mechanics 101 could you explain?

[@parasghumare8067]

Check my reply on pinned comment. You are right.

[@chenghui620001]

Thank you for this very impressive and informative video. I think there is a small typo in Eq (30): the normal stresses on the left side should be -puiui :)

[@fluidmechanics101]

Hmmm yes, I suppose this is a typo. As I am trying to combine the equations together I have left this expressed as -puiuj but you could also write -puiui if you wanted to be clearer. Good to see you are paying close attention 😂

[@ShamJam85]

at 27:33 you mentioned normal stresses for generalized equation but you have written it for shear stress since ui'uj' form. I am quite confused that this is in conflict to your earlier derivation for normal stress components.

[@parasghumare8067]

He wrote it that way so that he can make a single equation form in the end using Kronecker delta.

[@martinleskovec430]

Thanks for this. Could you do a video on turbulence models that are not eddy viscosity models, like cubic k-epsilon?

[@alexandergillespie6381]

This was really great, thanks.

[@TheAmitsch]

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?

[@Luiferhoyos]

Yes. A typo.

[@deepbandivadekar1974]

Stumbled upon this channel from a LinkedIn post. Incredible stuff. I have a question with regards to the illustrations shown in all of your videos, which are consistent in form/style. How do you create those? It clearly is some matpltlib-like library that has brilliant way to simplify this. Can you please share?

[@fluidmechanics101]

Of course. I use inkscape to make all my figures, being careful to keep the font consistent with the slides. I then import the figures as PDFs and don't resize them (to avoid stretching). Hopefully you can use this yourself? Inkscape is free and works on all operating systems, it's really good!

[@deepbandivadekar1974]

@@fluidmechanics101 fantastic, thank you! I am a proponent of OS packages. I already have inkscape but never realised its potential for illustrations. I'll definitely try.

[@pawejedrejko7398]

Actually, the deviatoric part of rate of strain tensor has a meaning. It's more often mentioned in the context of the von Misses hipothesis in strength of materials/elasticity theory (of course, in these fields, the strain tensor instead of the rate of strain tensor is used). It represents only this part of (rate of) deformation of infinitesimal fluid percel that changes its shape (changes the angles). The remaining rest, represents the 'resizing' of the percel - change of its volume (btw. thats why they sum up to divergence) without affecting its shape. It can be shown that pure 'resize' is represented by the average of the diagonal terms of velocity gradient. In 3D thats (du/dx+dv/dy+dw/dz)/3 or in 2D: (du/dx+dv/dy)/2. When you subtract it from the rate of strain tensor, then voila, you have the deviatoric part.

[@fluidmechanics101]

Amazing explanation! Much appreciated

[@TheClassicknowlty]

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!

[@fluidmechanics101]

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 👍

[@TheClassicknowlty]

@@fluidmechanics101 thank you!