[CFD] The Courant (CFL) Number

Рет қаралды 48,512

Күн бұрын

An introduction to the Courant number in modern Computational Fluid Dynamics (CFD) codes. The following topics are covered:
1) 2:07 How is the Courant Number calculated in 1D?
2) 5:41 How is the Courant Number calculated for general 3D cells?
3) 17:21 What are the recommended values of Courant Number in CFD?
4) 19:51 How does adjustable time stepping work?
#CourantNumber #fluidmechanics101 #cfd
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Some useful references:
1) The Original Paper by Courant, Friedrichs and Lewy (1928). The paper is in German so may be difficult to read!
www.digizeitschriften.de/dms/i...
2) CFD Online Discussion of Courant Number in OpenFOAM
www.cfd-online.com/Forums/ope...
3) CFD Online Discussion of Adjustable Time Stepping
www.cfd-online.com/Forums/ope...
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Want to learn more?
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Grab a copy of my CFD Fundamentals Course (for beginners):
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Learn how to write your own CFD code in MATLAB and python (for intermediates):
dr-aidan-wimshurst-s-school.t...
Learn how I draw my figures and diagrams in Inkscape (for everyone):
dr-aidan-wimshurst-s-school.t...
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www.fluidmechanics101.com/pag...
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Support the channel on Patreon (and get useful extras for your CFD studies):
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Disclaimer
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The methods, algorithms, equations, formulae, diagrams and explanations in this talk are for educational and demonstrative purposes only. They should never be used to analyse, design, accredit or validate real scientific / engineering / mathematical structures and flow systems. For such applications, appropriate trained, qualified and accredited (SQEP) engineers / scientists should be consulted along with the appropriate documentation, procedures and engineering standards. Furthermore, the information contained within this talk has not been verified, peer reviewed or checked in any way and is likely to contain several errors. It is therefore not appropriate to use this talk itself (or any of the algorithms, equations, formulae, diagrams and explanations contained within this talk) as an academic or technical reference. The reader should consult the original references and follow the verification and validation processes adopted by your company / institution when carrying out engineering calculations and analyses. Fluid Mechanics 101 and Dr. Aidan Wimshurst are not accountable or liable in any form for the use or misuse of the information contained in this talk beyond the specific educational and demonstrative purposes for which it was intended.

Пікірлер: 143

@maged779 3 жыл бұрын

What is amazing is that I have very little knowledge in CFD, yet I was able to fully understand this video and your explanation. Well done, really! Good teacher!

@eb89367 3 жыл бұрын

This guy is a gift.

@TheAnonJohn 2 жыл бұрын

great video. I love how you combine theoretical knowledge with its application to cfd codes.

@BiplabAdhikary 3 жыл бұрын

This is a real gem. The best channel for Fluid Mechanics. Thank you very much. ❤️

@PeterJang 6 ай бұрын

I learned CFD on the job. So many of these topics, including CFL (Co #), was explained to me in not such easy manner. You however explained it so well and helped me so much. Keep up the great work!

@winstonong9593 2 жыл бұрын

A Clear, Concise, and Comprehensive explanation of the Courant number. Bravo!

@mauriciorey9558 2 жыл бұрын

What a great explanation for this topic, I was able to understand every second of this video. Hopefully, all cfd books were as understandable as this. Thank you.

@upekaweerasekara332 3 жыл бұрын

I honestly haven’t seen anything more valuable than these set of videos in whole internet. I just don’t know where to find all these complicated theories this easily in one place. Your service must be immensely appreciated 👌

@gryphonflamingo141 2 жыл бұрын

Best lecture of courant number I've ever seen.

@manusego 2 жыл бұрын

Lot of thanks, teacher. Great explanation! I LOVE the way you clarify difficult concepts👏💞👏💞👏💞👏💞

@fahadaslam2011 3 жыл бұрын

Absolutely amazing, very well explained 👏🏻👏🏻

@MalcolmAkner 3 жыл бұрын

Great lecture as always, now I have a great grasp of the Courant number!

@marianzastawny156 3 жыл бұрын

Really nice video. Excellent job in explaining what CFL is. I think the video would benefit from a sequel, where difference of CFL in steady and unsteady problems is discussed. Also there is a difference in implicit and explicit methods which impact the stability.

@vivekbharti12 3 жыл бұрын

Brilliant explanation. Hats off to your efforts.

@lagoonbiswal9673 3 жыл бұрын

Absolutely flawless presentation. Thanks a lot sir

@jesuspereira4233 3 жыл бұрын

This lecture was great! I´m currently learning more about transient and steady state CFD simulations, and I did not know the meaning of the Courant number that well. This video was very helpful, I could said that it is better that some of the university courses that I have taken. Congrats and thanks for the help!

@martinagotti1660 5 ай бұрын

You have a gift in explaining! Thank you so much!

@yasiruranaweera3888 3 жыл бұрын

That absolutely a great presentation sir

@nileshgaikwad5021 3 жыл бұрын

Great explanation and presentation. Thanks.

@upsceshort 3 жыл бұрын

Really its very helpful for CFD users👍👍👍

@aryanmishra5591 3 жыл бұрын

You are simply the best. Thank you!!!

@316FOCK 3 жыл бұрын

Thanks Man! I see you from Mexico! Congrats for your very useful videos!

@manishankaryadav7307 2 жыл бұрын

Thank you again for the quality content.

@user-bf7iz2tz1i Жыл бұрын

Great explanation, thank you a lot! Very quality lesson.

@sriharisha251 2 жыл бұрын

extremely useful! amazing explanator

@ravikarantripathi7228 2 жыл бұрын

Really very knowledgeable video. Thank you very much, we are learning a lot from you.

@amirhoseingolpeykarrad1721 3 жыл бұрын

It is a very important lesson.

@febriyanprayoga7332 3 жыл бұрын

amazing knowledge Sir, thank you!

@karthikeyanpalani9387 3 жыл бұрын

Fantastic Lecture.... Thanks!

@jaewooklee3494 3 жыл бұрын

This lecture was great help for me!! Thank you so much :)

@SKKarthick12 3 жыл бұрын

Wonderful presentation!

@Ayush.Bhagat 3 жыл бұрын

Much needed information. Thanks

@itssachink 3 жыл бұрын

Thanks Man.. You are Great! Much need this one.😊❤

@shubhamphysicist 3 жыл бұрын

Clean and precise

@cronos864 Жыл бұрын

Thanks for the incredible video

@pauljnellissery7096 6 күн бұрын

What a lecture!

@jloudefonty3231 3 жыл бұрын

Clearly explained, thank you

@dr.gordontaub1702 3 жыл бұрын

Very nice presentation as always. Thank you.

@fluidmechanics101 3 жыл бұрын

Thanks Gordon 😊

@aravindm8512 3 жыл бұрын

God level Explanation

@mechtorious 3 ай бұрын

Very very good content. Thankyou.

@waynezeeman8948 3 жыл бұрын

Brilliant presentation...thanks

@TheDaniel96314 3 жыл бұрын

thank u, gretings from mexico

@giabu6287 3 жыл бұрын

You are saving my life

@maysamir8106 2 жыл бұрын

you are great teacher .thanks a lot

@phutanesiddheshrajan2253 2 жыл бұрын

Wonderful presentation ..

@JohnVKaravitis 3 жыл бұрын

Great video.

@sheetanshudeepak8779 3 жыл бұрын

Thank you sir, it was very good.

@usamaniaz2981 3 жыл бұрын

I always thought how courant no. Is calculated in 3D, now I know.. Thanks to you Aidan.. Impressive talk. I have a request, can you make a video on relaxation factors which we use OpenFOAM?

@childhoodforever223 8 ай бұрын

Thank you so much sir 🙏

@junjungarcia7236 3 жыл бұрын

great job sir...

@FILYPTC 3 жыл бұрын

thank you so much bro!

@nageshjamge1372 2 жыл бұрын

Great, thank again..

@phdmeme 3 жыл бұрын

Thank you very much!

@ozyilmaz4269 3 жыл бұрын

Thank you sir.

@prabup3114 3 жыл бұрын

Thanks Aiden..

@rwadalshebl7740 2 жыл бұрын

Thank you very much :)

@sarangpakankar6498 3 жыл бұрын

Hello Dr. Aidan. All your videos are extremely knowledgeable. Many of the concepts which were not clearly understood during graduation got cleared by your way of in depth explanation and use of extremely simple language. I just wanted to request you that if you could make one or two videos for important Dimensionless Numbers in CFD it will be of great help. Your simplicity of explanation will give us more insight into it. Thank you.

@fluidmechanics101 3 жыл бұрын

Thats a great idea, thanks 😊

@zaidarch 3 жыл бұрын

Thank u very much!

@mehrdadkhezrian3332 3 жыл бұрын

Awesome video! Please go to the population balance and polydisperse flow for the next video!

@RahulKumar-yk3kx 3 жыл бұрын

Thankyou very much

@ryszardb4653 3 жыл бұрын

Very usefull and easy to understand interesting form of presentation. But I miss some information about optimal and desired value of Co. Maybe some future movie? I appreciate what you are doing.

@VijayaLakshmiToed 10 ай бұрын

Hi,your lecture series are great. Please tell that whether courant number criteria has to be satisfied even for implicit solver as it is stable at any given conditions?Thank you in advance.

@SagarKumar-ks8me 11 ай бұрын

Excellent delivery! This happens when you know the shit you talk about. Thanks man.

@Jialei-dw3li Жыл бұрын

Dear Sir, I have a question: since the implicit temporal scheme is unconditionally stable, I can choose a large time step so that CFL can be larger than 1, am I right? In such a case, are there any constraints on the time step setting? Thank you.

@ashutoshsingh-et7vm 3 жыл бұрын

Great video sir waiting for your videos like any tv series but sir still waiting for blockbuster on large eddy simulation

@fluidmechanics101 3 жыл бұрын

... im working on the LES video right now 😄 it should be out in a few weeks time!

@egetopuzlar7836 3 жыл бұрын

Hi Aidan, I follow all of your contents. Thank you for your effort. If you inform us about Adjoint solver we will be quite appreciated. Respects...

@mounarahal256 2 жыл бұрын

Hello, do you mean by velocity fields and courant number fields the fact of plotting contours in fluent (for example for the courant number they are the coutours of cell convective courant number? thanks

@fluidmechanics101 2 жыл бұрын

Yes 👍

@ankitjaiswal3824 3 жыл бұрын

Thanks for this great video. Got to learn the basic understanding of the Courant number! I just had a doubt though: Do we need a courant number steady state CFD simulations too? Asking this I see the default Courant number listed as 5 in ANSYS Fluent for a steady state compressible flow simulation that I am doing.

@fluidmechanics101 3 жыл бұрын

For a steady state solver that uses 'pseudo time stepping' you can use whatever courant number you want, as you are trying to get to a steady state solution. However, you might find the solver diverges if you set the courant number (for the pseudo time step) too high. 5 is probably a good place to start. If it diverges, try reducing it a bit

@ankitjaiswal3824 3 жыл бұрын

@@fluidmechanics101 Yes, indeed, I am trying to run a simulation of the compressible flow through an Aerospike nozzle, and my results diverge for a Courant Number of 5. Also I happened to watch your video on 'Pressure inlet' BC. What should I take as the supersonic/ initial gauge pressure? For the compressible flow through an Aerospike that I simulating from a paper, the inlet Mach Number is close to zero in the contour plot. How close should I choose the supersonic/ initial gauge pressure with respect to stagnation pressure? I was wondering if a large deviation between those two can lead to divergence?

@racc2218 2 жыл бұрын

Thanks for give a such great explanation for CFL number. I have a question about the CFL

@fluidmechanics101 2 жыл бұрын

I think you can probably just ignore the first few time steps,as the transient is normally just washing out the initial condition then anyway 👍

@doemaeries Жыл бұрын

So far so good. But if I set Fluent to Density-Based with absolute velocity formulation and implicit formulation under "solution methods" the courant numbe ris set to 200 as default value. How can this work when it's generally recommended to be bolow 1?

@fluidmechanics101 Жыл бұрын

I think fluent sets the default to 200 for steady state simulations because it is using a pseudo transient formulation and can use larger than physical time steps. If you switch to transient, you can then bring the courant number down to 1 or lower

@totaloverdose3591 3 жыл бұрын

Sir how do we define courant no for multiphase flows with perspective of courant no as fraction of cell travelled ?Are they distinct to each phase or sum up as a single unit?

@fluidmechanics101 3 жыл бұрын

It depends if you are using homogenous multiphase or inhomogenous. If you have a shared velocity field (homogenous) then the courant number if for both phases. If you have inhomogenous then you can have a courant number for each phase 👍

@totaloverdose3591 3 жыл бұрын

@@fluidmechanics101 Thank you sir 👍

@najikajkolaji Жыл бұрын

great.thanks for sharing such a useful information.i have a fsi simulation in fluent part the courant number in controls section is 200 by default as my simulation is transient should i change the courant number below 1? or not?

@fluidmechanics101 Жыл бұрын

Good question. Maybe give it a go. Does it affect your results?

@najikajkolaji Жыл бұрын

@@fluidmechanics101 it took so much time to solve! i set the setting back to defalt!

@aidealczar6075 3 жыл бұрын

Hi Aidan, In the case of an airfoil, can it be assumed that as long as the lift and drag monitors are reporting constant values and the max courant number in the domain is less than 1 therefore the solution is somehow accurate?

@fluidmechanics101 3 жыл бұрын

... assuming your mesh is converged and you chosen the right turbulence model then yes 😄 to be sure of accuracy, you need to compare to experimental data

@aidealczar6075 3 жыл бұрын

@@fluidmechanics101 Thanks!

@mathanrajk6781 5 ай бұрын

Hi! How to calculate Del.x in CFL number in ANSYS Fluent?

@fluidmechanics101 5 ай бұрын

I think you can output CFL directly as a field, so I don't think you need to define del x

@user-vk1oh9pq2e 3 жыл бұрын

Is the value less than 80 not an error in the calculation recommendations? That is, if I simulate a centrifugal fan, then I can specify the values 79 78, etc. I don't quite understand why why such large values. Thank you for the video :)

@fluidmechanics101 3 жыл бұрын

A number of around 80 is just from personal experience with OpenFOAM and the pimpleDyMFoam solver. The max value you can get away with will be different for every CFD code. I would just try some different values and see what you can get away with. I used 80 as an indicator as i havent ever really see anyone be able to get higher than 80 without it diverging ....

@user-vk1oh9pq2e 3 жыл бұрын

@@fluidmechanics101 Thanks for the answer. I will try to use your advice and solve the problem for different values of the courant number. Will try to set an automatic time step in the solver and seen what happens. Perhaps this will increase the stability of the solution when using the k-e turbulence model.

@ConnorSmithBirch 2 жыл бұрын

So in Ansys, when we specify a Courant number, we are actually telling it the maximum number after which to not shorten the time steps?

@fluidmechanics101 2 жыл бұрын

I can't say for sure (as the source code is proprietary) but the idea is that the code will reduce the time step until the Courant number (either maximum or average) matches the value you specify

@ConnorSmithBirch 2 жыл бұрын

@@fluidmechanics101 awesome, thankyou!

@SwissCowboy87 3 жыл бұрын

Yesssssssssssssssss!!!!!!

@ashutoshsingh-et7vm 3 жыл бұрын

Great sir I was simulating jet impingement cooling on flatplate and consider delt t 10(-6) and max cohrent no 0.8 but my delta t reduce to 10^(-16) so i did many things it not works then I change discretion scheme of div (u) gauss cubic from gauss linear delta t adjusted to 5e-6 and cohrent no 0.24 sir does this affects my physics any way I am doing LES

@user-vk1oh9pq2e 3 жыл бұрын

I guess not, the models remain the same

@fluidmechanics101 3 жыл бұрын

Ahh yes there could be a number of things going on here. Have you checked your mesh and run a precursor RANS simulation? Try plotting the courant number and see where it is going so high. It sounds like you have adjustable time stepping, thats why the time step is going down so much

@sinangoren4803 3 жыл бұрын

ı understood that if the cfl number is higher than 1, it's already exceed the mesh that we are observing. But in turbomachinery you r reccomendations is less than 80. with that way how we gonna sure that these numbers are good for our cfd. I mean what might be coused if our turbomachinery cfl number is about 100. btw awesome presentation, thank you aidan :)

@fluidmechanics101 3 жыл бұрын

Yes, i probably should have explained this a bit better. In an ideal world we would always have CFL < 1. However for turbo-machinery in particular, this can be realllly slow ... so to get around this we use larger time steps (CFL > 1) and do more convergence within a time step. Really the solution is more of a series of steady state soluttions rather than a true transient. I have found from personal experience that you can often push up CFL as high as 80 without it diverging. The best thing to do is experiment for yourself 👍

@soroushasadian9100 Жыл бұрын

This video concerns transient analysis. I remember when I was trying to model a steady state supersonic flow over a wedge, I used the density-based solver, and in the control panel, I set the Co number to 5 according to the tutorial. Is it the same idea or not?

@fluidmechanics101 Жыл бұрын

Slightly different. I think the solver was using a pseudo-transient to arrive at the steady state solution (a pseudo transient is similar to a transient but does not fully resolve each time step and can use different time steps in each cell). The Courant number restrictions on a pseudo transient can often be less strict than a true transient, because we only care about the final steady-state solution and not on how we get there. As long as the solver doesn't diverge, the Courant number doesn't matter for pseudo transients. I have a lecture series on pseudo transients if you would like to learn more

@soroushasadian9100 Жыл бұрын

@@fluidmechanics101 Thank you, actually I recently watched the first part of the pseduo transient series. I thought it might be the time step in pseudo but I wasn't sure since you didn't mention it in the video Thanks again.

@hughtong5551 Жыл бұрын

I just confused by the factor 0.5 at 12:40 used by OpenFOAM and only when I watch this video I get to understand finally.

@TheAnthony761 3 жыл бұрын

Hello Dr. Wimshurst. First, thank you for this presentation, explanations are very good and efficient. I would like to be sure, this stability criteria of Co < 1 is only if you solve your equations with explicit formulation and it depends also of the numerical methods we are using in the code? I mean, in some situation, even with an explicit time integration, is it possible to use Co > 1? Also, for implicit time integration, I know that there is no restrictif criteria for Co, and we can normaly use Co > 1. I'm using FLUENT right now, but I don't really understand why with implicit formulation, with a great Co, I don't have stability and my calculation diverged.

@fluidmechanics101 3 жыл бұрын

Hi Anthony, it sounds like your simulation may have diverged for another reason. Have you checked: 1) your mesh, 2) your boundary conditions ? These are often the main causes of divergence. Try solving a simpler case with the same mesh to see if you can get convergence. This will help you identify why your simulation is diverging

@TheAnthony761 3 жыл бұрын

@@fluidmechanics101 I am trying to simulation the flow through a multi-stage centrifugal compressor and I am using the Mixing Plane Model (MPM) with mass-averaging method. The mesh is created by TurboGrid and is very good by default and the boundary conditions I have tried are Mass Flow Inlet / Pressure Outlet. I have tried also to perform with the Pressure (stagnation) Inlet / Pressure Outlet, but without success in multi-stage, when I include the MPM. Otherwise, with only one compressor stage (without the MPM model), it works...

@RicardoCruz-by7lv 3 жыл бұрын

I don't see physical meaning for such a huge diference for Co < 80 for turbomachinery in relation to the much, much, smaller figures of other common flows. Could you explain that?

@fluidmechanics101 3 жыл бұрын

Ah yes, i probably should have explained this more. With turbo machinery it quickly becomes very expensive to use small time steps as you often need to run several full rotations of the rotor to converge the solution. Hence, it is common to use bigger time steps (so that Co is >> 1) and converge more within each time step. I have found that you can often get away with Co up to around 80 before the solution diverges. I wouldn’t stand by these values though, they are just indicative that people often go for bigger time steps with turbomachinery. I hope this helps!

@dr.gordontaub1702 3 жыл бұрын

Not sure if I am confusing this answer by putting in my two cents or not, as it has been sometime since I have written my own CFD codes and started watching these videos because I would like to get back into it. But as I recall from reading Patankar's book which introduces the SIMPLE and SIMPLER algorithms, you can sometimes get away with your code converging even with higher CFL numbers if you use a method that is more implicate than it is explicit. Although I may be confusing two different issues. As I said, it has been a long time since I read this. I was also curious about the very high CFL number listed in the video for turbomachinary, so I thank you for asking the question.

@yusufpolat13 2 жыл бұрын

perfekt....

@ramgupta7697 3 жыл бұрын

I am facing negative cell volume in ansys fluent what will be the possible solutions for this and in fluent I am unable to find courant number option for pressure based solver (and my mesh is unstructured-triangular). Any kind of reply is helpful, thanks in advance

@fluidmechanics101 3 жыл бұрын

Hi there. If you have negative volumes you need to go back to your mesh generator and fix the mesh. Fluent cant fix these by itself, you need to go back to the mesh generator

@ramgupta7697 3 жыл бұрын

@@fluidmechanics101 thanks for your reply. Exactly, what should I fix? Should I increase the mesh size ?

@fluidmechanics101 3 жыл бұрын

Have a look and try and find where in the mesh the cells are with the negative volume. Once you find them you need to change the mesh structure (move the boundaries, blocks, or faces) so that the cells arent inverted. You can probably find them by looking at any quality metric (skewness, non orthogonality or determinant). They should all show bad values where the cells have negative volume

@ramgupta7697 3 жыл бұрын

@@fluidmechanics101 I checked the skewness, element quality, jacobian, and aspect ratio that seem in the range. Ok I will check other parameters too btw my geometry is 2D airfoil. Thank you so much for instant reply.

@axelcohen8795 11 ай бұрын

Very nice and physical explanation, I really enjoyed it. However, as I recall, the importance of the CFL number was mainly due to the method for time discretization. We can't go further than 1 for explicit scheme, however the value can be much much higher for implicit scheme. Am I right ?

@fluidmechanics101 11 ай бұрын

Yes correct!

@axelcohen8795 11 ай бұрын

Thanks ! So why LES recommandation for CFL should be < 1 ? It doesn't only depends on the time scheme (I mean, if it is implicit we could go further without worring to much about stability) ? Thanks again to revive all my old courses :D @@fluidmechanics101

@fluidmechanics101 11 ай бұрын

As well as stability, we also need accuracy, which is particularly important for LES. This is why we normally insist on y+ < 1

@hassanmohammadi2586 8 ай бұрын

and what about pseudo Timestep in steady state Simulation. btw it was really good

@fluidmechanics101 8 ай бұрын

I have a nice video on pseudo time steps you could check out 👍 I'm sure it will answer your questions

@gokaygunduzalp270 3 жыл бұрын

Thanks as a hec-ras user...

@vm1887 3 жыл бұрын

Great lecture. I have a question. The slide 'Stabilty' at 19:28 says criterion

@fluidmechanics101 3 жыл бұрын

To be honest, the criteria for turbomachinery is based on my experience with OpenFOAM. I couldnt really get a Courant number higher than 80 without it diverging. This will be different for different CFD codes and for different geometry. (Also my mesh was really high quality structured mesh, which might have helped)

@CarlosTapiaMan 2 жыл бұрын

How is the Courant number relevant for steady flows?

@fluidmechanics101 2 жыл бұрын

Only if the solver uses a 'pseudo transient' approach to achieve the steady state solution, you might need to reduce the size of the 'pseudo timestep'

@CarlosTapiaMan 2 жыл бұрын

@@fluidmechanics101 I see, I'll check out pseudo-transient flows. Thanks for the quick reply.

@CarlosTapiaMan 2 жыл бұрын

@@fluidmechanics101 As a follow up question, I've seen some Ansys tutorials of steady flow where they emphasize the Courant number setting. Usually I've seen c=5. (These are for steady compressible supersonic flows for example). I assume then this step doesn't really influence the result, since there is no flow "moving". Does that make sense? Thank you! Edit: spelling

@fluidmechanics101 2 жыл бұрын

Yes correct! If your solution is steady it does not matter how you get there as long as the calculation does not diverge along the way

@CarlosTapiaMan 2 жыл бұрын

@@fluidmechanics101 great thank you :)

@pattimichellesheaffer103 3 жыл бұрын

GREAT series of videos! Can you say something about "stiff" interphase transport? I believe this shows up as ODE source/sink terms in a set of flow equations (one for each phase). Does this constitute and "orthogonal space" somehow - that is, can CPUs be profitably dedicated to the solution of stiff ODEs? (an example might be OF reactingFoam) Thanks!

@fluidmechanics101 3 жыл бұрын

Im not very familiar with this area but i suspect it may be due to the interface compression algorithms. These often require Courant numbers < 1 to remain stable. Sorry i cant help you out with this one

@muhammadyasinshaikh5949 5 ай бұрын

how to get acees to cfd course for free?

@MaximumMatador 3 жыл бұрын

Found this because all Thermodynamic revision seems to be done by Indian professors, which is great if you can focus your full attention on it, or speak Hindi. I wanted to do revision in the background while I serviced my bike, so I decided to review fluid mechanics instead. A bit odd, really...